Bank limit beginning to rankle (4)
I believe the answer is:
tier
'bank' is the definition.
(both can mean a row of similar things)
'limit beginning to rankle' is the wordplay.
'limit' becomes 'tie' (tying is a kind of limiting).
'beginning to' indicates taking the first letters.
The first letter of 'rankle' is 'r'.
'tie'+'r'='TIER'
(Other definitions for tier that I've seen before include "Row in theatre" , "Person joining" , "Wedding cake layer" , "Layer, rank" , "Rite about layer" .)