Essentially, you can tune each individual string one step (2 frets) higher, with maaaaybe one more fret of tightness available. Open C tuning, for instance, is gCGCE, so the 1st string is a whole tone higher. Now, Appalachian players do tune a whole tone higher across the board (C --> D, G --> A, for instance), but I don't think you'd get away with tuning to open D (aDADF#). That 1st string would snap in a second. And yes, I know there's another open D tuning (f#DF#AD), but I was envisioning open C tuned to D.
What can help is to have a good wrap of string around the tuning posts. Some people put the string through the hole in the post and tune it with no slack, so that the kink where the edge of the post meets the string is less than two turns around the post. Not enough. It'll do for a guitar 6th string, but not a banjo string. Leave enough slack when stringing so that you get at least four (4) windings of string around the post when it's up to pitch. Better even would be 6, especially on the shinner strings (1st & 5th). When you put tension on a string, it pulls against the weakest point, the kink where the formerly-straight string gets wound around the post. If there is not enough friction from winding around the post, the tension is focused on the kink, which is the weakest part of the string. Give it lots of help. It only takes a few extra seconds to use enough wingings, and you won't begrudge the time when you have to replace a broken string.
So up two frets is a safe option, and careful winding of the strings will protect you from a nasty ping! when tuning.