Since DBs do not reuse numbers of deleted records it is possible to run out of numbers, especially if you pick not really a big integer type for this column.
What would
It depends on your database, I believe in MS SqlServer, you simply cannot insert any new rows until you fix the problem. The last time I encountered it, we fixed the problem by reseeding the identity column to 1. That's obviously not a universal fix, but it was ok with our situation.