When can the output of any flip flop (e.g., JK FF) be indeterminate?
Considering any kind of flip flop, yes, there are possibly indeterminate states at the output. The named example, level-triggered JK flip flop, might start to oscillate if the "forbidden" input combination is set. It depends on the technology the circuit is based on, the propagation delay of its gates, the exact timing of all input signals, and so on.

Another possible effect is metastability. This can stay for a indeterminate duration, and even worse produces illegal values "between" 1 and 0. After this unknown time it can finally set on a legal value which is indeterminate; it can be 1 or 0.