Why won't my recessed lights illuminate when they have power?
Evidence would suggest there is a wiring error. Recheck the wiring and all the connections - something is not connected, or is connected incorrectly. 

Since you claim to have tested that there is power to all the fixtures, one of two things is probably the case:


There is a high resistance connection which allows you to see voltage with a tester, but which causes the voltage to drop drastically as soon as a load is applied.
You have managed to get both hot and neutral hot, so voltage is present, but there's no voltage across the devices, as they are connected to hot on both sides.


As you might imagine, it's difficult to troubleshoot these things from afar. Doublecheck everything.