I only recently discovered that you actually can!
I visited wap.iguanamobile.com using my 7650's wap browser, went to animated pictures and saved one. It was stored as a file without an extension, yet it animates all right! I event sent it in an mms to a 3510, which stored it as an animated picture and now can be used as an animated screesaver.
The animation itself can be viewed in the built-in "images" application as well. There's also AniViewer (freeware, about 10K) which loops animations.
I only confirmed that you can and gave an example how to download some. I found it interesting that you can send them to e.g. 3510 via mms. Is there another way to send them from the 7650 to phones such as 3330 which do not support mms, though? Via sms maybe?