Which real-life musical groups have names based on puns?
The prototypical example would be The Beatles (beat / beetles).
Novelty, parody, and cover bands tend to do this a lot, as with Dread Zeppelin (who play reggae renditions of Led Zeppelin tunes) and Petty Theft (a Tom Petty tribute band). But I’d really like to hear some punny names for serious recording artists that put out original material.
The Surfaris.
Like the fictional Oneders, I think a “serious” name is usually imposed by a band’s first manager or record label. Lucky for the Beatles, “The Beetles” was even ickier.
Camper Van Beethoven
Def Leppard
Led Zeppelin…maybe. Jimmy page thought it was a pun on “lead balloon” but he may have had the same English teacher as Alanis Morissette.
Me First & the Gimme Gimmes
Indie bands used to live doing this. Off the top of my head, there’s Japancakes and The Dandy Warhols.
Concrete Blonde
Dead Kennedys
Where is the line between puns and word play?
There are the Brian Jonestown Massacre which is the latter I think.
There is also Pete Doherty’s solo project the Babyshambles
A light orchestra is a group somewhat smaller than a symphony orchestra, so Electric Light Orchestra is a twist on light orchestra, most of their instrumentation is electric and throw in a light bulb motif and you have chuckles a-plenty. 
As for a cover bands that’s a thread-worthy topic by itself. More punny ones off the top of my head:
AC/DShe (All female AC/DC cover band)
Abbaesque
Rolling Stoners
Was wondering the same. Heavens forbid I list a name and have it not make the cut 
AlicebanD is both Essex-based frontwoman Alice Walker’s project and a British term for a horseshoe-shaped hair accessory
Regrettes is a twist on the existing word, pronounced the same but visually indicating that it’s a group of women
Prettiots and Shebrews are both word play by merging stuff.
Teen Jesus and the Jean Teasers is more just alliterative wordplay than playing with word meanings.
Also all reinforcing InternetLegend’s comment that indie acts are chock-a-block with this nonsense.
Did they “lift” that from (Buddy Holly and) The Crickets?
Hayseed Dixie (originally an AC/DC bluegrass cover band)
There was a local cover band called The Next Doors. The lead singer stayed in character the entire time. He was Jim Morrison.
Lunachicks
Jack Off Jill
I don’t get the pun.
WAG-it’s a play on “platinum blonde”. I could be wrong.
Yes, they wanted a name evocative of Buddy Holly and the Crickets.
Thanks.