Vocal Range Exceeded

Gags where someone comically tries and fails to sing a note too high or too low for him or her.

Examples of Vocal Range Exceeded include:

Music

Parodied in "The Dooright Family" by Ray Stevens, a song about a fictional gospel singing family. At the end, the bass vocalist is asked to go down an octave, which causes a huge, loud, booming note that makes him explode on stage.
While producing The Temptations' "Ain't Too Proud to Beg", Norman Whitfield deliberately arranged David Ruffin's lead vocal just above his actual range, requiring numerous takes to get all the high notes right and adding to the mood.
Note that straining one's voice too much will result in a sore throat.
Parodied by Mozart in his composition A Musical Joke not with vocals, but the violin. At the end of the violin solo near the end of the 3rd movement, the soloist plays ascending scales, slowing down and playing a whole-tone scale at the higher registers. This was to imitate a clumsy violinist floundering with high notes. The solo can be heard here (with the ascending scales starting at 5:12).
David Gilmour admitted he had trouble reaching a high note in Pink Floyd's "Welcome to the Machine", so they resorted to recording that note a semitone lower and then playing it back at the right pitch.
Non-comedy example: Neil Young's "Mellow My Mind", where he's simply too tired, too drunk and too sad to reach the highest notes. The effect is rather heartbreaking.
A part of "Alto's Lament". Although I've got a great high C... *cue this trope*

Theatre

At the end of "Do, Re, Mi" from The Sound of Music, Maria sings a scale starting "do, ti, la" from middle C, and can't quite make the final "do". (This gag was not used in the movie because Julie Andrews had a higher vocal range.)
Spamalot does it in "The Song that Goes Like This".

Arthur: (squeakily) Now we're into E!
(clears throat) That's awfully high for me
Lady of the Lake: As everyone can see
We should have stayed in D...

Web Original

Oancitizen does this in his musical review of The Man Who Fell to Earth.

Western Animation

In "Long-Haired Hare" ~~Leopold~~ Bugs Bunny directs pompous opera singer Giovanni Jones to sing a note far below his vocal range.

Other

"Open Sesame Seeds" from PDQ Bach's oratorio The Seasonings gives the bass a ridiculous melisma that descends lower than he can actually sing.
Anna Russell's coloratura aria parodies "Canto dolciamente Pipo" and "O gentle bird with feathered breast" end with cadenzas that are obviously going to end on notes high above the staff, except that, after a few seconds of breathing, she instead sings her final note two or three octaves lower.
Regularly used on Svengoolie when his musical director Doug Graves arranges songs for Sven to sing, just a bit higher than Sven can sing.
Common advice to people learning Arabic in pronouncing the letter `Ayn is to "sing the lowest note you can, then one lower." This leads to all kinds of amusing sounds until the student finally gets it right.
This is entirely possible when playing vocals in Rock Band, to the amusement of anyone else in the room.
Miss Piggy gets into a vocal range duel on Ride of the Valkyries with a real opera singer (Beverly Sills) at 1:55 in this video. She can't quite hit the high note.