Almost broke case from Norway that's thin (6)
I believe the answer is:
skinny
'thin' is the definition.
(similar in meaning)
'almost broke case from norway' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'broke' becomes 'skint' (both can mean struggling financially).
'case from' suggests removing the centre (the case or outside of the word).
'norway' with its middle removed is 'ny'.
'skint' with its last letter taken off is 'skin'.
'skin'+'ny'='SKINNY'
'that's' acts as a link.
(Other definitions for skinny that I've seen before include "Very slim" , "Of person, very thin" , "little of substance?" , "Scraggy, scrawny" , "Low-calorie white coffee -- sent any kilt (anag)" .)