Whether Bing can compete with Google on a technical level is up for debate. If they say their server farms, software platforms and indexing is as good as Google's, it should be fairly easy to test.
But from a user standpoint, Bing is probably just as good (or better) than Google. The average user uses Google and/or Bing to search the Internet. The more relevant the search results, the happier the user. And from what I have seen, Bing's results are no worse than Google's.
We shouldn't think about this from a webmaster or SEO perspective. Webmasters may not like hearing it, but neither Google nor Bing sees webmasters and SEOs their primary concern. The average user is what concerns them. How much traffic they send you, or whether or not their ad-serving matches your page content perfectly comes second.
The bigger issue for Microsoft's Bing (and other search engines) is that Google has become synonymous with search. The phrase "Google it" is akin to the phrase "Xerox it," or other phrases that associate a business with a process. Google has held that position for years, and it will be difficult for another search engine to dislodge that association from the minds of average users.