This may only be a problem with my particular sound card, Envy24, but I wanted to pass this solution along in case anyone else was having it.
Normally, Skype works just fine but under some circumstances for which I have not discovered a pattern, it thinks the sound card is missing. The result is that as you click on "answer" to take an incoming call, the call is rejected and Skype tells you that "There's a problem with your sound card inside your computer".
If you get this problem, this fix is to turn off "Allow applications to take exclusive control of this device" under Sound/playback/speakers/advanced". You may have to do the same for the microphone as well. There's a more detailed description here: Skype sound card problems on Windows 7