Thank you for the information and the explanation. I appreciate your trouble. However, the mystery has just deepened.
I was discussing the problem with the person who gave me the computer, when she mentioned that she had another identical one! Apparently three were purchased as insurance replacements several years ago. So I was given the second Acer to experiment on. This computer appears to be identical in every respect to the first one. Model number and other specs are the same. The motherboard markings are also the same. The only difference is that the second computer has had a memory upgrade! Instead of the two 512 MB modules that were in the first one, it has one module of 2 GB and another of 1 GB. I am aware the Acer manual says this computer model only supports 2 GB of RAM, but that appears not to be the case. The computer I am having trouble with works fine with a 2 GB module in one slot as long as the 512 MB module is also present, as previously explained. Accord to the memory diagnostics I have run, as well as some applications, the extra memory is also actually being used.
The second computer also works fine with 3 GB of RAM. I have tested that. Yet if I take the exact same modules out of the second computer, and put them in the first, I get the same problem. The computer won't start. There is one slight difference. If I put the 1 GB module, which is actually an official Acer module, in the second slot (furthest from the CPU), then I do get an initial display from the BIOS, which immediately freezes. With any other RAM, or with this module in the first slot, I see nothing at all, no beep or other sign of life, just a completely blank screen.
Again, just to be clear, two upgrade modules (2 +1 GB) work as expected in computer # 2. They do not work at all in computer #1, except I get the beginning of the BIOS screen if the 1 GB module is in the second slot. Both modules work fine one at a time in either slot, as long as the 512 MB module is in the other slot, exactly the same as every other piece of RAM I have tried.
So what on earth is actually going on here? I am perplexed. Except for this stupid RAM issue, computer #1 seems to work perfectly in every other respect. I have been using it for several days with applications installed to test it. I am not a hardware expert or engineer, but it seems to me like this must have something to do with the motherboard. It just doesn't seem to make any sense.
To answer your question, I did check all the memory I was using in tests to see if it was the same frequency. As far as I could tell, they were all the same, though the designation was a little ambiguous on a couple of them. The 512 MB module that does work is marked PC2-4300, DDR2-533. My understanding is that Acer's 4300 is actually 4200.