GPS accuracy is largely a funciton of how many satellites your reciever can see. In deep valleys or behind large mountains the accuracy can degrade significantly. Most purpose built GPS units provide feedback in terms of what satellites they are using and/or what the accuracy of the displayed coordinates are.
When using UTM coordinates it is critical that you set the GPS to match the geodetic standard that was used in producing the map your are using - it should be printed on the map somewhere. In North America the two standard used by both the US and Canada in their topographical surveys are NAD27 (North American Datum 1927) and NAD83 (North American Datutum 1983). Think of the geodetic reference as the pattern that was followed peeling the rind off of a orange in order to lie it flat on a table. This is not an issue if your are using longditude and lattitude, but UTM coordinates are much easier to use for navigation on the ground as they give you measurements in meters. By just looking at your current location and knowing the UTM location of where you are heading you can say I need to go X meters N or S and Y meters E or W.
There is a movement to move toward the WGS84 geodetic standard which is very close to NAD83, but we still have to live with multiple standards for now - particularly in Europe where it seems every country has it's own standard. I have personally had major grief with this issue while trying to navigate through a small mountain pass while skiing across the French-Swiss border near Chamonix in near white-out conditions. I was using Swiss maps, we were still in France, I wasn't sure what geodetic standard was used for the coordinates that were in the guide book that I was referencing and visibility was about a half rope length (30m). Where the GPS was telling me the pass should be there was a 600 foot granite spire! You only let that happen to you once!
