My general answer is that rechargeables can be trusted just as much as an alkaline, but there are LOTS of qualifying caveats associated with that general answer. In some cases, Alkalines are going to be better, in others rechargeables. In MOST general applications though, I don't think either battery chemistry would give me any more or less concern over the other.
However, the more detailed answer is to basically agree with Jon's response above, except that I don't particularly agree that one battery is better or worse, even say if rechargeables are old and have higher internal resistance. IR isn't bad, in and of itself, as long as you recognize what it does to battery performance and deal with that in the application you're using the battery for. For example, it's possible that someone might want a battery with high IR for some reason.
Anyway, as Jon said, when you get into the particulars of battery performance, analyzing one chemistry against another can be quite complex. It's all a matter of considering the variables associated with the particular battery you're using (voltage delivered, internal resistance, battery calendar age, battery 'number of cycles' age, discharge curve, etc.) vs. the variables associated with the device being powered (voltage needed for power operation, current draw, cut out voltage, run times needed, continuous draw vs. momentary draw, etc.). Each of the battery variables may or may not have a desirable impact on the variables impacting end device performance. And the specific performance goals you have for the end device might be variable depending on the different ways you use the device.