Most smart cards have 'passive RFID' chips on them which don't require an on-board power source and thus do not have an actively transmitting antenna. They are usually powered by electromagnetic induction, i.e., placed closed to a reader that powers up the chip. Failure rates, thus, are low for passive RFID chips and more often than not a read failure occurs when the reader isn't able to pick up signals properly due to attenuation (signal weakening due to distance / material obstructions) rather than the chip on card itself failing. Other reasons could include extreme temperature variations that cause chip to contract/expand but operating range on most smart cards can work fine for -20 deg C to 50 deg C range.
TL;DR: There's no easy way to determine whether your card has failed (unless it's physically bent/damaged), but card failure is not THAT big a problem.
Most national ID smart cards also have additional security features such as holograms/watermarks to prevent counterfeits so border officials will still have a way of determining (to an extent) whether it's genuine. Even if one does fail, I'm sure immigration/border control authorities should have procedures in place, similar to those already in place for damaged paper passports.