I worked out part of our problem. We're using EQUELLA as our learning resource repository and the 'view' settings were set to HTML5. When we switched it to 'file viewer', the videos came up fine!
Other videos I uploaded still don't want to play, however.
- Firefox plays MOVs, but has a problem with WMVs and MP4s uploaded from my desktop.
- Chrome will play MOVs and MP4s, but also doesn't like WMVs.
- Internet Explorer doesn't want to play anything... Unless it's a video from EQUELLA, which is acceptable to us.
- Didn't test Safari, but an iPad reacted like Firefox.