In terms of probability:
Any part that directly generates heat (CPU, GPU, RAM) and any moving parts (PSU fan, CPU fan, GPU fan, mechanical hard drives) fail the most often
In experience
1. GPU's fail the most often, either the FAN fails or the RAM fails. The observable effects of both are "Windows has reset the graphics driver" and unexplainable graphical glitches, and Windows refusing to load Aero. When it hits a critical failure state, it simply BSOD's
2. Hard drives fail the next most often, and often the failure is progressive. Errors on mechanical hard drives are always caused by
a) Wear (I had two drives from the same manufacturer, fail within a week of each other, despite being in different PC's) those machines RAID systems were unrecoverable and warranted a reinstall.
b) Inappropriate shutdowns (pulling the plug, power switch, power spikes/sags/surges)
c) laptops and portable drives usually suffer from "drop damage"
3) PSU failure, I'll admit that I've replaced PSU's for people going back to the 90's and their symptoms are usually the computer works for a while and then BSOD's or shuts down/reboots. This is caused by voltages dropping for whatever reason. The second, and less obvious symptom is that the PC won't turn on, or gets stuck in a "POST loop", where it's drawing too much power on boot up. This is not to be confused with...
4) Motherboard failure, which has the same "POST" loop problem that may be a result of a firmware update gone bad. Clearing the CMOS will usually fix it, but I had one board (combined with the CPU/RAM timing problem) kill itself, despite having a rescue firmware.
5. CPU or RAM - Now the only way either of these fail is by thermal damage. It's far more likely that the CPU and RAM are a bad match than the parts independently going bad. This is usually why your Dell/HP/Gateway systems come with low-end slower parts, because they get a higher yield out of them, despite shorter lifespan.

The AMD 6700 series is not a "low end card", that's actually the 120$ price point card (the current model is the 7750) Switching the video card from one brand to another is usually not the correct answer since that just exchanges one set of driver problems for another.

My personal opinion is that the crashing sounds like the Motherboard (MSI is great and producing cheap, low-quality parts, I won't touch them) , also I would probably not recommend going to BestBuy for advice, as the people there (even if they aren't commission) are still primarily sales people, and their technicians are high-school kids. However the crashing may simply be motherboard settings, not defects. The "time till failure" mentioned is something that happens from bad CPU/RAM timings (I replaced every single part in my previous PC chasing down a similar problem, ultimately replacing the CPU fixed it, but that was because I went from a Xeon to a C2D chip that was 60% slower.)

The motherboard is supposed to select the most appropriate timings. However sometimes you have "high performance mode" or "turbo settings" set when you probably shouldn't.