Worked at a FedEx Kinko's for a couple years, including doing pack-and-ship for customers. Everything we packed had to be able to survive a three-foot fall from any angle. Even reasonably sturdy stuff got a lick of bubble wrap and a nice cushion of peanuts. Shipping an old CRT monitor? Bag it and tape the bag shut (just in case), wind it in bubble wrap 'til its 3" deep, stuff that into a box with peanuts, and then stuff THAT box into another box with at least 3" of peanuts. Those laptop kits they make everyone ship their computers in? When the first ones came in, my manager put her laptop in one, then let the staff kick it around the store like a soccer ball. Came out without a scratch and booted up just fine.
FedEx and UPS use a lot of automated processes to handle packages. These machines are only so gentle. Add the "accidents happen" factor, and, yes, you should make sure you package your items well. Sticking a "fragile" sticker on a flimsy cardboard box does not magically encase your hand-blown glass sculptures in a mystic bubble of protection; that's what the bubble wrap and peanuts are for. Use it, and use lots.