Yes, unfortunately, you can definitely overcharge Eneloop batteries. This is possible when you use a dumb charger (time-based charger). The charger will continue supplying a charging current for X number of hours. The battery charger will only stop when the timer has run out.
Another way is by trickle charging.
The most common problem is with using DECT phones that stand on a charging cradle most of the day. These charging devices provide a very low charge current to keep the batteries charged. The problem however is that Eneloop batteries don’t like this type of charging.
For Eneloop PRO batteries it is even disastrous because this will reduce their cycle life tremendously. This is because they are much more prone to heat and overcharging.
More questions about charging (Eneloop) batteries
Advanced Eneloop charging info
+ What is -dV/dt charging? (termination)