I've used this strategy in the past to get additional stay credits. It only works if you check into a different hotel on day 2 or if the reservations are in your name for day 1 and 3 while day 2 is in your wife's name.
I've tried separate reservations at the same hotel for 2 separate nights and it was 'fixed' at corporate level - loss of amenity points and only counted as 1 stay for 2 nights.

I'm currently working on getting *Wood Plat status back (they won't comp it because I've been a Plat in the past). This last weekend, I stayed 4 nights at 3 different hotels. Sheraton, Hyatt Regency, Sheraton, Element (in that order).
I got 3 nights + stay credits with *Wood. I got 1 night + stay credit with Hyatt.
If you want to get the points, it requires additional work on your part. I can't do that strategy when I'm staying with my wife; she hates switching hotels. Needless to say, my wife wasn't on the trip.