Pinch flats or debris.... A few things to check out are as follows (coming from someone that might have ran 35psi in 2.1" panaracer Fire XC):
-Tube size. Correct tube for tire? Better err on the smaller (width) size. Tubes stretch better than they fold over themselves in a tire!
-Rim tape. If you have a rubber rim strip, toss it and install some Velox rim tape. Best stuff out there.
-Sharp nipple holes/nipples poking through - de-burr before installing rim tape where necessary.
-Something is in your tire. Check the inside of the tire, make sure nothing sharp (rocks, glass, thorns, etc) is poking through, remove anything that is. Running a cotton ball along the inside of the tube is much safer than your finger. Anything sharp enough to damage a tube will grab some cotton and you will see it
-baby powder the inside of tires lightly so the tubes don't stick. This just makes life easy.
-inflate tires partway after installing (10-15psi should be enough), check around rim for tube sticking out between tire and rim, and if its ok, inflate fully to your desired PSI.
-MOST IMPORTANT: make sure tubes are not being pinched AS
you install them, between rim and tire. I sometimes have to use a tire lever to get a tire on, I am super careful if I have to. Try to only use tire lever to remove tires..
I don't even pump my tires much past 50 to ride on smooth fine gravel paths. 35psi offroad is about what I would use. 60 is WAY too high for offroad use (this is MY opinion anyway...)
If you still have problems, take the wheels to a bike shop and see if someone can help you out, or find a fellow mountain biker to maybe look things over?
That is all I have for now. Been a while since I have been real mountain biking, but I was pretty thorough with my tire/tube installation whenever tires/tubes were removed. Never had a flat, either. I almost think I am still riding on 8-10 year old tubes...