They have to rely on the Group D leaders, Kent, beating Sussex at Hove in a game to be completed today.
Although Marcus Trescothick hit 122 of Somerset's 316 for 5 and held four slip catches, it was Graham Rose who took the gold award after scoring 47 not out off 48 balls and taking the the top four Irish wickets for 21 runs.
The other eight ties were all disrupted by the weather, but not before Bill Athey wrote himself into the Sussex record books as his side made 303 for 6. Athey struck 118 - the highest score by a Sussex player in the competition - including 10 fours off 156 balls.
The Kent captain, Mark Benson, and Trevor Ward had put on 31 in seven overs when rain cut the day short.
Graham Gooch struck a superb unbeaten century as Essex reached 211 for 4 in 49 overs against Hampshire at Southampton in Group C. He was 117 not out when rain halted proceedings. His half-century came off 93 balls with his second 50 taking only 55 deliveries.
Nasser Hussain chipped in with 46, figuring in a century stand for the third wicket with Gooch. Keith Brown made 75 and the New Zealander Dion Nash 54 in a sixth-wicket stand of 119 as Middlesex made 276 for 8 against the Combined Universities at Lord's. The students had reached 135 for 4 in reply when play was halted.
The holders, Warwickshire, can qualify for the last eight if they beat Minor Counties and Nottinghamshire lose. The West Midlands side recovered from the loss of their opener Andy Moles with the score on one to reach 100 for 1 at Edgbaston.
Nick Knight (42no) and Dominic Ostler (54no) helped the home side reach three figures before the rain came after 23 overs.
Only 18 overs were possible in Lancashire's game against Durham at Old Trafford. Put into bat, Durham reached 57 for 2 with both wickets falling to the Pakistan bowler Wasim Akram.
Rain prevented any play in the match between Derbyshire and Yorkshire at Chesterfield.