Ignore all the naysayers, a 1000W PSU is BUILT to be run AT 1000W Output and the warranty covers any failures - most PSUs that high end come with 5+ years of warranty.
What you measure as 930-960 is at the wall or INPUT, so the output is most likely 930x0.9 or~840W - a perfectly safe number.
There are ofcource some other considerations like temperature (most PSU ratings are at 45-50C) and the input voltage - on 110v most PSU's get like 80% of peak rated power output. SO if youre running on 110v or in a very hot room with no cooling, better get a 1200-1300W PSU, otherwise a 1000W is just fine.