I've got a 2005 Gibson Les Paul Studio whose 4th (D) and 3rd (G) string saddles are slipping.

I noticed it not too long ago. As I tuned both strings, the tone would not change for a quarter of a turn and then jump past the note I was trying for. As an example: when tuning the 4th string, my tuner would read Db and as I moved the tuning machine sharp the note would not change for about a quarter of a turn and then the tone would jump past D and go to D#. When this occurs, I hear an audible "click" sound that originates near the bridge. This is not a tuning machine issue: I already replaced the stock machines with better ones previously.

This issue has appeared after multiple string changes. I finally took the Tune-o-matic bridge off the guitar and noticed that the 4th and 3rd string saddles wobble. I can put my fingertip on those saddles and physically move them back and forth with little pressure while the other saddles do not move under similar pressure. I believe that this is the cause of my problem.

I have already changed out the stock tuning machines and toggle switch previously and was told by an expert that the stock Gibson hardware was "crap". I tried tightening and loosening the saddle threads but the saddles still wobble back and forth. Is this just another instance of "crap" hardware and I should accept the fact that I need a new bridge? If I do need a new bridge, what are some good suggestions?

Any feedback or suggestions would be GREATLY appreciated!
Saddles "wobbling" isn't your problem, it's the nut. My saddles wobble like crazy on my tune-o-matic, but it's much more likely the string catching in the nut. You hear the click by the bridge simply because the string suddenly jumps a small amount, which could either momentarily release the pressure on the saddle and be quickly reapplied, sort of making the saddle do a small jump and landing. Or some other variation of that.

Easy enough to test though. Try using some graphite to lubricate your nut the next time you change strings. If that doesn't seem to work, get a nut file and carefully widen the slots on the nut for the affected strings by a small amount. If that doesn't work, then maybe it really is the bridge after all, though I really doubt it.

Regardless, it's not that big of an issue. Stock Gibson tuners are typically what, grovers? Grovers are pretty solid machines, so I don't know why you replaced them. It sounds like your "expert" just wants your money, to be honest.
It could well be the bridge and if so just buy a better one, you can get them from places like axesrus in the uk but maybe somewhere like stewmac might have them in the US. More commonly the sound/problem you're having is actually the nut, the sound travels up the string so it sounds like the bridge. Check how loose or tight the stings are in the nut slots and maybe try lubing them with something like a pencil or nut sauce.
Thanks for the quick replies guys!

The reason I replaced my tuners was that one machine's case began prying off the machine and tuning became near impossible. This was my decision and not my expert's.

I will be changing my strings here soon so I'll let you guys know if lubrication works.
I slightly enlarged the slots and lubed them up with a tip of a pencil and everything seems to be working flawlessly! I'm surprised that this isn't better posted on blogs and such seeing as how easy it is. Thanks again to everyone!