The most common and most likely answer to this crossword clue is the 5 letter word LISPS
I've seen this crossword clue in The Daily Telegraph
L
I
S
P
S
More clues leading to the result LISPS
Buddy explains!

The straight part
We know that "Says I’m thick, when I’m not well" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case says i’m thick, when i’m not well = lisps.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 LISPS mean?
LISPS noun- The habit or an act of lisping.
- To pronounce the consonant ‘s’ imperfectly; to give ‘s’ and ‘z’ the sounds of ‘th’ (/θ/). This is a speech impediment common among children.
- To speak with imperfect articulation; to mispronounce, such as a child learning to talk.
- To speak hesitatingly and with a low voice, as if afraid.
- To express by the use of simple, childlike language.
- To speak with reserve or concealment; to utter timidly or confidentially.