One of the first occurrences we came across was John Cleese, who developed a type of nonsense English and has used it to great comic effect. Have a listen to this interview on NPR, around the 8:50 mark, where he talks about this nonsense language he developed. He's also used it relatively recently:
The interesting thing about this clip is the confusion it brings about - it sounds and feels like understandable language, but is in fact complete nonsense. I've passed this on to a few people I know and some have reported that it makes them feel uncomfortable as it sounds like it should make sense but doesn't.
In the NPR interview linked above, John Cleese states Stanley Unwin and his Unwinese as his inspiration for his nonsense language. I hadn't known about Stanley Unwin before, and it turns out he's absolute master of gibberish - and very, very funny to boot:
It's also possible to speak grammatically correct English without making an any sense either. The late great Ronnie Barker delighted in messing around with pronunciation and spelling, to great comic effect:
Ronnie Barker and the News at Ton.
And now for the final variation: grammatically correct, valid English that unintentionally doesn't make any sense. For example, see this classic spoof of the Turbo-Encabulator:
I think it's interesting that this low signal-to-noise-ratio style of speaking is instantly recognisable to anyone who has encountered people who speak a lot of needless business jargon. Observe this unintentionally hilarious clip from Microsoft COO, Kevin Turner, weighing in at 4 minutes 20 seconds:
If you take that same clip, and reduce it down to just business buzzwords, its 1 minute 49 seconds long - that's 42% buzzwords(!!):
I'll just leave this here: