Can I charge and discharge a NiCd/NiMh battery at the same time?
No, you cannot simultaneously charge and discharge a battery. The net current through it will do one or the other, but not both.

The real question is, can the charger simultaneously charge the battery and power the load? That depends on the design of the charger and the nature of the load, neither of which you have provided.

You seem to be concerned about high-current short-duration loads, in which the battery would switch from charging to discharging and back again. This should work fine, as long as the charger itself is properly current-limited in order to prevent damage to it. When the load switches on, the battery will make up whatever current is needed beyond what the charger can supply.

Also, such load impulses may confuse the end-of-charge logic in the charger (assuming it has that kind of "smarts"), so you'll have to take that into account in the design of the overall system.