Just about all RTC chips require a 32.768khz crystal for timing reference.
The 2 small caps are definately required and some of the chips are a little fussy about the value of the caps.
I wouldn't think these parts would go faulty, other than being broken on the board by mishandling etc.
What's the number of the RTC chip? if we dig up a data sheet, we may find there's a few pins that provide other timing signals which are measurable with a probe/meter. (1 pulse-per-second output is a typical signal).
I guess the next step is to determine which pin(s) carry the data to the main CPU, and confirm thsoe tracks aren't broken anywhere, maybe a dodgy connector?
Are the volts to the chip ok? 5V stable etc etc?
MM.