If you can put a CRO somewhere in the oscillator section and try tapping the crystal - you may have an old fashioned dry joint. Once I cured a colour TV problem by re-soldering the actual crystal. RTCs can suffer from stuck bits, but it does not seem likely in your case. Rubbish spikes on the power lines could reset the clock*, are all your input lines dealt with properly? - no floating ones?
Frank
* Just thought when all the bits reset from 235959 to 000000, the device itself might take a pulse of current and dropping the supply - have you local decoupling caps?