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I applied for an RA position (there are two posts). The professor responded promptly and with much enthusiasm.
Then, no more responses for two days. I just got an email from him apologizing for getting back to me late. He said there was more response than expected, and he would have to wait till next week to make decisions.

Have I lost already? How can I possibly boost my chances?

If the prof is not scheduling interviews, then stop by the prof's office on your own. Reiterate your interest in person and have a casual conversation with him (doesn't have to be about the position in particular, just something to get to know one another as people). That will help you stand out a bit.

RA spots are hardly strictly meritorious. A lot of profs drag feet because they see their position as a job of last resort, and so don't want to take somebody then have them get an offer with a judge, firm, PI shop, etc.

I wouldn't read much into the prof's email. Reiterate your interest, and keep looking for other jobs. You can't ever let yourself get too attached to one position in job hunts, the odds are always against you being 'the one'.