Almost tense when one's asleep? (5)
I believe the answer is:
night
'when one's asleep?' is the definition.
(people sleep at night)
'almost tense' is the wordplay.
'almost' becomes 'nigh' (nigh can mean almost or nearly).
'tense' becomes 't' (grammatical abbreviation).
'nigh'+'t'='NIGHT'
(Other definitions for night that I've seen before include "Thing (anag)" , "''Tender is the ...'' (F Scott Fitzgerald)|" , "follow-up to 16 [TWILIGHT]" , "Dark period" , "See 1 Across" .)