Serious but not grave (5)
I believe the answer is:
acute
This might be a double definition.
'serious' is the first definition.
(acute can mean serious in a medical sense)
'not grave' is the second definition.
'acute' can be an answer for 'grave' (I've seen this before). I'm not certain of the 'not' bit.
'but' is the link.
Can you help me to learn more?
(Other definitions for acute that I've seen before include "Shrewd; severe" , "not seen in cafe here!" , "Severe, penetrating" , "Very perceptive, having sharp mind" , "Critical, serious" .)