Sure, but FDIC insurance is per depositor, per insurable account category, per institution. Someone with three different savings accounts at the same institution each with less than $250k individually but more than $250k total still only gets $250k in insurance.

EDIT: I'm not saying the system can't, for a lack of a better word, be gamed, but ultimately that requires action on the depositors end, not necessarily the bank(s).

EDIT 2: Which means that any balance over that 250k in the same category would, in theory, count towards an institution's uninsured deposit balance.