Victory against their Group B rivals at Trent Bridge enhances third-placed Gloucestershire's chances of reaching the knockout stages of the county competition.
Hammond hit 22 fours and five sixes in his 157 with Bancroft contributing 52. Ollie Price (61 not out) eased Gloucestershire home with 20 balls to spare.
Nottinghamshire skipper Haseeb Hameed had earlier made 105 and Lyndon James 61 as the hosts were bowled out for 332 in 49.3 overs.
Somerset moved two points clear at the top of Group A after a five-wicket win over second-placed Worcestershire at Taunton.
Lewis Goldsworthy struck 95 and James Rew 70 as Somerset passed their target with over five overs to spare, reaching 5-267 after bowling out Worcestershire for 263.
Tom Taylor boosted Worcestershire's below-par batting display with a 65-ball 73 from No.7, while Somerset all-rounder Ben Green claimed 3-58.
Kent dented Derbyshire's bid for a quarter-final spot with a three-wicket win in Derby thanks to captain Jack Leaning's unbeaten 81.
Leaning and Grant Stewart (48) helped Kent to 7-207 in the 46th over after Derbyshire had been bowled out for 206 despite wicketkeeper Brooke Guest's 85.
Also in Group A, Colin Ackermann was run out four short of a century as Durham damaged Hampshire's top-three hopes with a convincing 144-run victory at Gosforth.
Ackermann made 96 off 98 deliveries in Durham's 257 all out before Hampshire were skittled for 113 in reply, with Paul Coughlin taking 3-41 and two wickets each for Neil Wagner, Bas De Leede and George Drissell.
