It's unclear what you mean by "run-of-the-mill internet downloads", because that implies there's some sort of reliability about them. Nothing could be further from the truth - you are stuck with the way that others have encoded it. If the files are XviD or DivX then especially packed bitstream, Qpel, GMC and suchlike can mess up the playback if you have an outdated decoder.
Like JayC30, I'm against the idea of codec packs. You never know what you're getting in them, and the wrong sourcing of the pack may include hacked versions that breaks DirectShow (the 'standard' relationship between the elements of the decoder train that you need to play them back).
I would second reasonsnotrules suggestion of Gspot. As she says, this app will tell you if your DirectShow can put together a filtergraph to play the A/V, but if not then it will tell you what codecs are needed. Then look for these from reputable sources - DivX and XviD decoders you can get free for a start.
If it's an .avi, a further step is to see what's 'included' in your downloads' encoding strategy - the site of MPEG4 Modifier will give you some clues about the Qpel, GMC etc, and a nice application to remove packed bitstream without re-encoding.