I know someone who did a PhD and MLS concurrently. The PhD program knew about the MLS degree, but he did it at a different school. Considering the state of the job market right now, it's really not a terrible idea. My program actually had an info session for PhD students on earning an MLS, because it might be helpful in the job search.
I definitely would give yourself the first year or two to just focus on your PhD program, to get a sense of the expectations and work load. I would also suggest you look for an inexpensive option, because Library degrees really are not as name-conscious as PhD programs. My friend who did both really regretted the $$ he invested to go to a top program, because he realized he could have gone to a lower ranked program for less $$ and the same benefits. Also, here are a number of reputable online programs (which are based in a brick-and-mortar school, so we're not talking UPheonix here) that could really help you with the time-management side of things.
On one level, it's a crazy things to do, but then isn't he PhD program in and of itself a little nutty? Be careful that you don't bight off more than you can chew with coursework or finances.