1) Disable the Theme Developer module if you have it installed
2) Click the "Hide Row Weights" link to remove the weights column from the display
3) Go to Configuration > Performance and turn off all the cache and optimization settings
Yes, I've had that happen on a couple of sites for me also.
I've had to rely on the weights feature to re-arrange the fields.
There's a lot of people with the same problem here drupal.org/node/487934 and there's a variety of suggested fixes:
- replace the "/misc/tabledrag.js" file with a fresh version from the latest Drupal project download
- disable the Devel module