Sometimes, although other animals are worse.
The textbook case is a mare who is about to foal who doesn't seem to realize her foal isn't there yet and tries to take somebody else's. More rarely, a maiden mare will try to steal a foal. The only solution is to separate the animals - it can result in fights between the mares, the foal getting hurt, or the "victim" mare losing her milk.
A mare who has lost her foal will also sometimes try to steal another mare's foals - a behavior pattern that can be made use of if you have an orphaned foal.
I've also heard stories about donkeys trying to steal both foals and calves!