The "bad" about that is that if "to google" became generic for "doing a websearch" is to stop other search enginge to start saying "Google it with Bing!".
I would have thought that would be "good" for brands to be so firmly entrenched into society that the brandname becomes a verb. "Google it with Bing!" would lead to the thought: "No, I'll use the real thing."
Imagine the benefits to Bing if Google were to use "Bing it with Google."