One problem with Dremel tools is that they don't always age gracefully, after a number of years of service they can develop a lot of lateral play in the spindle. Since the majority of uses for a Dremel place a side load on the spindle, it deforms the bearing mounts over time, I'll show you a quick trick to tighten them back up.
This Dremel I picked up at a garage sale had almost 1/16 of an inch of visible lateral play before.
Step 1: Crack it open
Pretty simple, unscrew the annulus behind the collet and remove the four T15 screws. Pull apart the two case halves, remove the spindle/armature, and it's as apart as it needs to be.