It sounds like it's suffering from brownout; ie, the output voltage "droops" briefly to a level where the processor's behaviour (and probably other components, too) becomes indeterminate.
In this situation, it is very likely that a simple RC circuit will not properly reset the processor when the power recovers - so anything could happen next! (like the software locking-up).

Therefore, the first thing you need to do is to ensure that you have a 100% reliable reset mechanism - use a dedicated supervisor chip.
As I said before, this does not fix the problem - but it ensures that you can recover reliably from the problem.

Other problems might be noise coming in through the power supply (or other connections), or grouding (earthing) problems. Again, a Supervisor circuit with Watchdog will not cure these problems - but it will ensure (if correctly used) that you can recover reliably from them.

Now you need to review your power supply design:
+ Is it adequate to fully power you circuit through all possible input surges, spikes, droops, etc?
+ Is it (and all other connections) adequately filtered?
+ Is your grounding good?