Do errors in Memtest86+ definitively mean something is wrong with my RAM?

I recently built a new computer for somebody. Everything seemed fine at first, but when I went to install the OS (tried both Windows XP and Windows 7), both installers would fail. XP failed with a BSOD, Windows 7 just said it could not read some file and the disk may be corrupt, although one time it BSOD'd as well.

I decided the next logical step would be to run Memtest86+ on it, so I booted into it and let it run for a while. I watched it for about 5 minutes with no errors reported, but I came back 45 minutes to an hour later and it was reporting over 15,000 errors. I opened it back up and removed/reseated the RAM. I even put them in the other set of DIMMs (dual channel with 4 DIMMs total).

When it came back up, Windows 7 installed and seemingly ran correctly. I thought "problem solved", but now, a day or so later, the system is back to BSODing. I have yet to run Memtest86+ on it again (haven't had time), but I suspect the same thing will happen.

Is it safe to say that the RAM I bought is faulty? Are there any other tests I should do to make sure that's the problem?

It's fairly safe to say that there is a hardware fault when MemTest86 fails. RAM is the most likely culprit, but there are a few other things that it could be in rare circumstances. I'll list them in decreasing likelihood:

1. RAM - Yes, the RAM could just be bad. This is the answer at least 95% of the time.
2. Power Supply - RAM operation is fairly sensitive to power fluctuations. An ailing PS could have all sorts of odd effects. The only real test is to swap in an known good PS with lots of extra capacity.
3. Motherboard - This connects all of these components together, leading to a lot of possible failure points. A motherboard problem is just as hard to diagnose as a power supply, though.
4. CPU - You'd probably see other problems if your CPU was having trouble. That said, modern CPUs have memory controllers built in, so memory errors could be a result of a problem CPU. Running a CPU stress test is a fairly reliable way to ferret out these issues.

Memtest86 can also report errors if the memory timings in the BIOS are wrong. This can happen if you've modified the default timings, or if the SPD values embedded in the modules are incorrect, or if the BIOS is interpreting the SPD values wrong. I once had the latter occur. A BIOS update fixed the problem, so I suggest you try that before swapping out hardware.

It's also possible that there's a compatibility problem between components in the system, where nothing in isolation is malfunctioning in other systems, but all your particular pieces fail when put together.