These batteries are down to 1.2v around 30-50% of the way through their life.
The rechargeable batteries start around 1.2v, but their discharge curve is much more level. They stay near 1.2v right up to the end.
If a device has trouble with rechargeable batteries because of the voltage, that device is also going to have problems with non-rechargeables, as the non-rechargeables spend at least half their service life below 1.2v.
There are some good graphs showing discharge curves here: http://www.powerstream.com/AA-tests.htm
The rechargeable batteries can confuse battery meters on some devices, though. If the device assumes non-rechargeable batteries, and then tries to estimate remaining battery by comparing the voltage to a typical non-rechanrgeable discharge curve, and you put rechargeable batteries in, the device is going to think they are starting out at 50-80%, and then is going to think they are draining very very slowly.
A good example being the Microsoft Bluetooth Mobile Keyboard 6000. The batteries don't last for 5 minutes even. Alkaline batteries last for months.