I'll throw in my 2 cents. Misson essentially said everything you need to know, but I'll add some details I've used to track down issues in some very difficult to find RAM errors.
I would test all the memory. Emachines uses "economical" components that don't typically hold up as well as some of the more expensive computers.
RAM can be quirky and difficult to test. I recommend the following:
Find a program called memtest and make a bootable media. A CD-ROM is the most likely to work without messing around. There are tons of different versions under a ton of different names, and they all work fairly well. Make sure your version is bootable, not a version that runs in windows.
Label one stick of RAM A and the other B.
Label one DIMM (RAM slot on the motherboard) 1 and the other 2.
Run memtest until it completes 2 full passes with the following setups (stop testing as soon as you find an error):
A in 1, 2 is empty
1 is empty, A in 2
B in 1, 2 is empty
1 is empty, B in 2
A in 1, B in 2
B in 1, A in 2
Certain combinations of RAM and DIMM slots will show memory errors and some will not. If you find any of these configurations cause the computer to fail the bios check or fail to boot (the computer will beep and display an error, the screen will be blank and the computer will beep, or the screen will be blank and nothing will happen when you power on) then all sticks in the computer are bad. If memtest has any errors, all the sticks in the computer are bad.
Some experts recommend using two sticks of the same RAM, so if one is bad, you may want to consider buying two sticks. I'm not an expert, but that's my recommendation. I'd also recommend Mushkin, Crucial, Patriot or Kingston RAM from
http://newegg.com.
Poor grounding and static discharge are things that have been proven to damage electronics in certain scenarios, but I think it's more likely that either the system wasn't fully discharged (the computer was turned off, but the power supply was still plugged in or the power supply capacitors had not fully drained), the stress of removing and putting the RAM in the DIMM damaged a connection on the RAM stick, or the RAM is not seated in the DIMM correctly. Don't be too shy when putting the RAM in the DIMM. It should and will take a little force to do. I usually open the little "wings" that hold the RAM in place, seat the RAM in the DIMM, and push down with one finger on each corner of the top of the RAM. The "wings" will snap into place and lock the RAM in the DIMM if everything is OK.
Lastly, since it's an old computer and it wasn't an especially good computer when it was new, it may be time to get a new one. Try seeing if there is a college in your area. A lot of college IT departments buy computers every year and gradually replace all the computers on campus. The old computers often get sold off for less than $100 (my college sold them for $25) and they are often high quality machines less than 2-3 years old. They will have no operating system (Windows will not be installed and the license key sticker will be scraped off the side of the chassis), but you can always throw linux on there. Big businesses also do the same thing. Some local community schools do as well.
TLDR; Memtest it. Maybe get a new one. Linux.