Staying dry is important when camping or hiking at high elevations. The threat of hypothermia is always present, all the body needs to do is drop its core temperature below 35C (95F) and you could be in trouble. Even without risking hypothermia, being wet is no fun time.
BRING RAIN GEAR
Sounds simple but a lot of people, backpackers and day hikers alike, simply don't pack rain gear. Even when the day calls for sunny skies, remember it can rain in the mountains almost anytime. A ranger in the Smoky Mountains told me it rains somewhere there every day.
If you are day hiking or light weight backpacking bring along a wind shirt and wind pants made with EPIC fabric. EPIC fabric is super light and resists water enough that it can be used in all but the most extreme conditions. It's perfect for those fast moving afternoon thunderstorms and rain showers that last up to a few hours at a time. EPIC is not a DWR fabric but instead the fabric itself is encapsulated with silicone, making it a good wind and water blocker. Two other benefits of carrying an EPIC wind shirt and wind pants are the EPIC fabric dries quickly and can be used to stay warm in high winds even if it doesn't rain. On the downside EPIC is breathable but not as much as with other materials.
If you expect heavy rains, snow or extreme condensation I recommend using rain gear made with eVent. I used Gore-tex jackets and pants for over 10 years, and while Gore-tex performs well and has its definite uses in one years time I've totally become a believer in eVent. eVent sheds water more efficiently than Gore-tex leaving the garment lighter and drier feeling. I wore my eVent jacket and pants during 9 straight hours of rain and the water continued to bead up on the jacket and pants the whole time. Gore-tex by contrast seems to soak up the moisture after several hours, becoming heavy and moist feeling. eVent also breathes, blocks wind better and dries faster than Gore-tex. In the snow the fabric seems to tighten up keeping me dryer and warmer than Gore-tex. This is just from my personal experience, as always use whatever works best for your circumstances. I'm only giving suggestions, do some research, test gear and develop your own rain gear systems that fit your uses and environment.
WEAR CLOTHES THAT DRY QUICK
This is another sort of obvious but overlooked aspect of staying dry. In summer if you do get wet (while wearing rain gear or not) and you have quick drying clothes on, when the rain quits, with the right conditions such as sun and/or wind, you can dry out in10 to 15 minutes. I wear any type of T-shirt or base layer that wicks away moisture. I also wear underwear or bottom base layers that wick moisture. In summer I prefer to wear quick drying shorts that have a sewn in net type underwear support or else I just go without. If you must wear long pants they can be found in quick drying materials too. By doing research you'll find that stuff made to dry quick comes in all sorts of weights, from light to heavy depending on the amount of heat/cold you will encounter. In winter wear DWR fabric pants or fleece pants. There are also pants made for winter that have a DWR fabric outer layer with a synthetic insulating layer like Primaloft for example. The key here is these synthetics will keep you warm when wet, are lighter when wet and dry faster when wet. You can also get a synthetic pullover insulated with Primaloft to use as a mid layer in winter. In cold conditions your shell will be keeping out the bulk of the rain/snow but if any does get through the pullover will keep you warm even when wet. The socks you wear should also be of the type that wick away moisture and if you know your going into a wet situation always carry extra socks. Wool socks will wick moisture and keep wet feet warm the best, many wool/synthetic blends do quite well too. For wearing on your feet, trail runners and modern Gore-tex hiking boots or boots with synthetic fabrics are all made to dry quickly in the sun or wind. For example I've been able to dry my wet trail runners while wearing them on my feet just from a combination of sun and wind and from my moving around in them. This won't always necessarily work unless conditions are right; for the record it was on the first day of November and I'm not sure how well it would work further into winter. Of course when it's warm you could hike in sandals which dry quickly. Sometimes I do it based on terrain and load weight. Sandals are something to definitely think about on a creek or waterfall hike, along with the possibility of stream crossings the terrain may be quite suitable to wearing sandals .
KEEPING YOUR SLEEPING BAG DRY
A dry nights sleep is a good nights sleep. If you have a down sleeping bag I suggest ditching the manufacturer's stuff sack and using one that is waterproof. If you get caught in a storm with no pack cover or fall in a creek soaking your backpack your down bag will be saved. Many down bags now come with DWR fabrics for a shell. Some even use waterproof fabrics like eVent and Gore-tex, DriZone or Pertex for their shells. These water shedding or waterproof fabrics can help keep the down inside your bag from becoming wet during periods of heavy condensation inside a tent or if your staying under a tarp and catching some splash water or wind blown drops. A bivysack made from DWR or waterproof fabrics can help tarp campers keep stray water from finding its way inside a sleeping bag. I use an Oware DWR bivysack when ever I'm tarp camping, it works well against all forms of moisture including dew which accumulates practically every morning year round in the mountains here in the southeastern US.
I recommend a synthetic sleeping bag in all seasons except the coldest months of winter. If your winters are wet or you expect to be in extreme conditions go ahead with synthetics year round. In summer I use a Mountain Hardwear Lamina 45 bag and for early spring and late fall I use the Lamina 20. Synthetic bags are easier to keep dry than down. The synthetic fabrics in a Mountain Hardwear Lamina sleeping bag will dry in the sun or wind in an hour or two (in my experience). Its possible to lay inside a synthetic bag even if your wet or the bag is wet and the insulating properties are not compromised. You will stay warmer and drier than in a wet down bag. This subject is controversial; ask one hundred backpackers which is better down or synthetic and it will turn into a slug fest full of insults to each others intelligence. I'm just telling you my opinion based on thirty years of experience. In two years I've pushed my synthetic bags to extremes I never would do with down and now the last twenty eight years of using a down bag seem archaic to me. Remember though, in winter (December - February) I still use down, but synthetics are way ahead of down when it comes to getting and staying dry.
HOW TO GET DRY
If it's winter always carry at least an extra baselayer in case you get wet. You'll at least be able to get warm dry clothes onto your core which is very important, then you can begin drying out the rest of your stuff if you are able. Go ahead and carry extra clothes if you want, depending on how light you want to go you can choose what extra stuff to bring. In summer for example I carry no extra clothes, just what I have on and my EPIC wind/rain shirt and pants, winter varies depending on temperatures and snow/rain outlooks. Carry a light weight but strong cord you can use as a clothesline, you can even use the same line you use to hang your bear bag in an emergency but try to have at least a four foot dedicated line for clothes or gear to dry on or use the lines on your tarp or tent to hang stuff on if they are long enough. When the rain stops you can hang all your wet gear to dry. Take advantage of a clothesline it is basically natures dryer. You can sleep with wet socks in between your bag and sleeping pad or bag and bivysack; sometimes you can put them inside the sleeping bag with yourself, but these tricks for wet socks don't always work. If some layers of your clothing get wet and you have extra layers either get out of the wet clothes and get them hanging to dry and simply change clothes, or you can put dry layers of clothing on so the wet layers are in between dry ones and then cover yourself with your sleeping bag. The heat created by your body will begin to escape and dry out your wet layers. I've used this method many times with great success. Again, this technique won't always work because conditions up to and including how wet and cold you really are will determine the actions you'll have take to dry yourself or face the possibility of things like freezing to death or simply staying up all night just plain wet and uncomfortable. Building a fire also remains a very effective way to dry out body and gear. Besides a warm sun and wind, fire is by far the best way to dry out. A fire is especially effective for drying boots and shoes. Learn to build a fire in any conditions. If you got to deal with wind, rain or snow where you're going you need to be able to build a fire in any of those conditions. Bring along more fire starting materials than the minimum, don't leave yourself short, without this essential ability you can be in trouble. Everyone should bring fire starting materials wether backpacking or day hiking.
Good luck and stay dry!