Demon's Souls has a maximum weight load for your inventory. As in, not the stuff you have equipped but the stuff you have in your backpack. It's not a bug, it's a mechanic that they rather wisely chose to not bring back for future games.
To increase this maximum load, raise your vitality stat. If there's anything you want to keep but don't want to carry around all the time, return to the Nexus and talk to Stockpile Thomas to store it. Note that if you attempted to pick anything up and were unable to, if you leave the area and return it'll be gone forever. If it's something you wanted or needed, it might be best to drop some other, less important items to make room for it.
If you talk to Stockpile Thomas and give him that (quest item I can't recall the name of, ?hairpin?), he'll give you a ring that boosts your inventory weight cap.
I have to say though, that system would probably have been a whole lot more interesting if recovery items weighed more than a tenth of a weight unit. Running around with two stacks of 99 full restores without any kind of penalty kind of made it a moot point.
Rest in pieces, Manus.
Useful notes for Dark Souls 2, by yours truly: