Yes, but are they on the same fuse as the front lights or other loads, where the additional 3 amps of the new lights pushes you over the fuse rating, but being just a little over, it took some time to blow the fuse?
Regarding using a bigger fuse: Fuses (and circuit breakers) do 2 things:
1) Instantly protects and isolates the "upstream" electrical system (alternator, battery, etc..) and all system wiring when you have a downstream short circuit (essentially this is remotely shorting the battery terminals together) and you maybe have 100's of amps now running through the wires in the instant before the fuse blows; and
2) Protecting the downstream wires from carrying too much (over)load current, overheating and melting. (Example: #16 Awg wires may only be rated for 10 amps so are protected by 10amp fuse).
With a bigger fuse, you will still have protection for short circuits, but maybe not for overloads on the wire, depending on it size. (Example: If you put 15amps load on a 10amp rated wire protected by a 20 amp fuse the wire might become its own "fuse" that melts and opens the circuit. Hopefully it doesn't do this next to something combustible). Check wire sizes before upsizing fuse. Realistically, wires can take a lot more current than their "rating" and not be damaged, they'll just create more heat than is wanted. Splices and junctions would suffer and tend to fail more from too much overload when the fuse doesn't blow.
I can't tell from your OP, but do your front lights still work? Did the fronts go out before you cycled the switch? And then the fronts came on but the backs stayed off?