Fellow responsible for locking up greatest possible lag (4,6)
I believe the answer is:
fall behind
'lag' is the definition.
(I know that lag can be written as fall behind)
'fellow responsible for locking up greatest possible' is the wordplay.
'fellow' becomes 'F' (abbreviation - of a society etc.).
'responsible for' becomes 'behind' (I've seen this before).
'locking up' indicates putting letters inside (I've seen this in other clues).
'greatest possible' becomes 'all' (I've seen this in another clue).
'f'+'behind'='fbehind'
'fbehind' enclosing 'all' is 'FALL BEHIND'.
(Another definition for fall behind that I've seen is " Lag".)