There are a couple of possibilities that could be causing this.
One is the bolts holding the wiper motor have worked themselves loose and simply need re-tightening. See
http://www.xoutpost.com/articles/x5/...wiper-fix.html
Another possibility is the wiper assembly has simply worn out and needs to be replaced. That is what happened on my 2001 X5 and is not as easy (or cheap) of a fix, but it is not beyond the skills of a good DIYer.
p.s. - there is a good reason why this only happens in bad weather - it's the only time that you use the wipers
Once the wiper motor binds, it will shut down for a period of time and then reset itself. Hence the reason you see them working again after letting them sit for a while.