cut it/things fine

Picture source: llow very little time for something

She arrived ten minutes before her flight, so she was cutting it a bit fine.

(Definition of cut it/things fine from the Cambridge Advanced Learner's Dictionary & Thesaurus © Cambridge University Press)

cut it (or things) fine

allow a very small margin of something, especially time:boys who have cut it rather fine are scuttling into chapel

Idioms & Phrases

cut it fine

Do something closely with a very slight margin, as in Tom always cut it fine, arriving at the last minute .This term uses fine  in the sense of "narrow." [Late 1800s]

The American Heritage® Dictionary of Idioms by Christine Ammer.
Copyright © 1997. Published by Houghton Mifflin. 

If you cut it fine, you only just manage to do something- at the very last moment. 'Cut things fine' is the same. 'Cut it a bit fine' is a common variation.
Read more at http://www.usingenglish.com/reference/idioms/cut+it+fine.html#gET0wdMuIrzFzVDp.99