Can you use the paired t-test in what seems like an unpaired t-test situation? I came across a situation where a large retailer is testing a new retail program.  It selected specific pilot retail stores to test this program.  Next, it selected a control group of retail stores so that it would be as similar as possible as the pilot retail stores.  It matched each pilot store to one control store based on numerous sociodemographic and business metrics.  The selection criteria was to generate a P value from a student t test as high as possible, indicating a good match between two stores.  This all made good sense.  On the other hand, they told me they used the paired t test.  This was a surprise.  I thought this would be a classic unpaired t test situation.      
 A: The use of matching on other characteristics than that being tested, followed by a paired t-test is fine; there's a point to that (to control for the effect of other variables thought to be relevant to the outcome). 
The matching makes it paired, because - if the characteristics on which the matching was carried out* affect the outcome - the pair-members will tend to be more alike than two random observations.
* or other characteristics which are related to them.
See, for example, the discussion in the third paragraph ('A paired samples t-test based on a "matched-pairs sample"...') here.
Henry's point that the pairs should be selected before deciding on which is to be the pilot (randomly) is a good one, though that problem won't necessarily invalidate the test - depending, for example, on how those pilots were chosen. 
However, unless Dimitriy's supposition was correct - that two lots of tests were done, one on the matching variables, and then a different test later for the effect of interest - then the way the matching was done, using the p-value of the test of interest as the matching criterion, renders the resulting p-value meaningless. It's certainly no longer uniformly distributed under the null. In that case, the hypothesis test is effectively useless as it stands.
