If anyone is still following this thread:
My previous responses were based on my memory so I decided to pull out an old friend, Malvino's book on Electronic Principles that I taught about 15 years ago. Aball's mention of re prime brings back fond memories.
On CE amps with the emitter bias resistor totally bypassed, the AC input signal is developed completely across re prime (the AC resistance of the base-emitter junction). Since re prime is affected by temperature, the gain of the amp is temperature dependent. Not good. To solve this, a swamping resistor is added in series with the emitter bias resistor that is not bypassed by the cap.
Which brings me back to my original statement about bypass caps NOT adding to the stability, the one before I rescinded it. The correct response is that bypass caps on a BJT amp actually make them less stable with temperature changes.
Clueless, you are absolutely correct. It seems everything that is done to a circuit effects it in ways other than the one intended. Adding a bypass cap as above to increase gain and bandwidth also increases distortion and decreases input impedance. Coupling caps do indeed affect frequency response and can affect the bias, especially on high impedance input circuits such as tubes by holding a charge when it was not intended to.
I enjoy these discussions and don't take any differences of opinion as arguing (except maybe on posts like the one on the Dixie Chicks). Unlike politics, electronics is based on the science of physics so for the most part there is very little that can be argued. It just that it is complicated enough that a lot people, including myself, get confused and develop strange ideas about how things work. Hopefully by sharing we can all get a little smarter.