Wanting to swap out the speakers in my Bugera 412 cab that's 120 watts. Would I be able to put 4 Vintage 30 Celestions in there? Since each one is 60 watts that'd be 340 instead of 120, can a cabinet's wattage be adjusted by just throwing speakers in or does it actually matter?
A cab's wattage is only adjusted by changing the speakers. A wooden box has no inherent wattage handling characteristics.

Your typo aside, your new cab would be 240 watts.
How you wire the speakers affects both load in ohms and power handling.
"Your sound is in your hands as much as anything. It's the way you pick, and the way you hold the guitar, more than it is the amp or the guitar you use." -- Stevie Ray Vaughan

"Anybody can play. The note is only 20 percent. The attitude of the motherfucker who plays it is 80 percent." -- Miles Davis

Guthrie on tone: https://www.youtube.com/watch?v=zmohdG9lLqY