I'm reading a bit between the lines here, so this may not apply. If it does apply, then this is the fastest and easiest way to debug a possible memory problem in this situation..
But...it SEEMS like you're saying that at least sometimes the computer does boot up into Windows and runs for some short period of time, before the failures start to occur. If that's not what you meant, if in fact the computer never passes Power On Self Test (POST) then the following does NOT apply.
IF that is the case, then we know your system can use the RAM. They may not be blessed by the motherboard manufacturer, but at least they are electrically compatible.
IF that is the case, then remove all RAM except 1 of the new sticks.
Make sure it is installed in the correct slot. Then reboot and try to run Windows again.
If it's stable, then that stick is probably ok.
REMOVE power again. Replace that 1 stick with the other new stick of RAM. Then restore power, reboot, and try to run Windows again.
If it's stable, then that stick is probably ok.
If either or both sticks failed, then it/they are bad or have a subtle compatibility problem with the motherboard, or your motherboard can't handle two sticks of RAM. If you never ran 2 sticks of RAM before, this is possibly a motherboard issue.
Replace the failed stick(s) if any. As kOrny indicated be sure to verify the memory you purchase is compatible in timings AND in voltage. For example, there are some memory sticks that take 1.65V, and your motherboard may be able only to supply 1.5V...which could lead to unstable memory...although I think given the memory type your motherboard uses you won't run into memory like that, that even physically fits in your memory slots. Not absolutely positive, just covering the possibility. I think all DDR"type1" RAM is 2.5-ish volts.
Remember to ENTIRELY remove power from the PC, then waiting for at least 30 seconds before changing parts. This means either using the PSU's back switch or removing the AC plug. Simply using the front PC button does NOT remove all power and you will likely damage parts by doing so. In fact, if you didn't remove power ENTIRELY when you first installed the RAM, it's possible you were the one that may have damaged the RAM.
Similarly, use good ESD practice when handling computer parts. While it may not cause any immediately observable problem, ESD can cause future part failures.
Source: http://www.hardwareanalysis.com/content/topic/77351/ (didn't have a lot of time so just copied)xD