Usage of IE should not cause such a problem.
Perhaps you have some other piece of software that has a memory leak.
I would suggest that you remove all of your cards with the exceptions of the graphics adapter and modem and see if it persists.
Also, remember that a single ctrl-alt-del brings up a small window that shows what is loaded into memory. That might prove informative. Better yet, if you have the Norton Utilities or Norton SystemWorks, bring up SysInfo click on the "memory" tab and see whats there and how much memory each is taking.
If all of this proves uninformative there is another alternative. Do a ctrl-alt-delete and "end task" on the topmost item and run for awhile and see if that changes the situation. Repeat this process over and over again until, hopefully, you find the culprit.
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Also, I am unaware of any Windows issues above 256MB of RAM. There is definitely one after you cross the 512MB boundary, but that should not be causing you problems with your 384MBs.
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My best wishes to you on a speedy resolution of your "disappearing memory" problem.