The most common and most likely answer to this crossword clue is the 5 plus 1 plus 5 letter word CHOCK A BLOCK
I've seen this crossword clue in Financial Times
C
H
O
C
K
A
B
L
O
C
K
Buddy explains!
The straight part
We know that "quite full" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quite full = chock a block.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CHOCK A BLOCK mean?
CHOCK A BLOCK adjective- (of a ship's hoisting tackle) Having the blocks drawn close together so no further movement is possible, as when the tackle is hauled to the utmost.
- (by extension) Jammed tightly together; very crowded; completely filled or stuffed.
- In a crowded manner; as completely or closely as possible.