Sure, under two conditions in each case:
- the conditions that made them extinct in the first place no longer prevail (i.e. we aren't bringing back species just to have them go extinct again a few generations down the line)
and more importantly
- the impact of their reintroduction can be modelled beforehand with sufficient accuracy that we can be confident that they will not collapse ecosystems or unbalance them enough to cause more than an expected rate of extinction
Those conditions would apply regardless of the original reason for extinction, which (clearly contrary to many other people in this topic) I see as utterly irrelevant. If it's good to bring them back, do it regardless of the reason they went extinct. If it's bad to bring them back, they stay extinct, even if it was entirely a human-caused extinction. If we can't tell with confidence whether it's a good or bad decision (and this is the most likely scenario for the foreseeable future), then we don't bring anything back.
Concerns about "playing god" are misfounded. Humanity having the ability to determine whether a species goes (or stays) extinct axiomatically means humanity has power at least as godlike as anything in most pantheons. Exercising that power by making a choice about bring back a species or not is "playing god" either way.