The only time we did not accept (since we started going to bowls) was when the Seniors voted to not go to the Independence Bowl in 1996.
It was the players who voted though, not the "school"
I don't believe '96 was the only year ND teams chose not to go to a bowl but it may be semantics over not accepting a particular bowl invite as opposed to any not accepting any bowl invite.
ND did not go to a bowl in '71 after a 8-2 season under Ara.
ND did not go to a bowl in '75 after a 8-3 season under Devine.
ND did not go to a bowl in '79 after a 7-4 season under Devine.
ND did not go to a bowl in '96 after a 8-3 season under Holtz.
After the 1925 Rose Bowl win the ND Administration decided not to participate in bowl games as they were exhibition games and interfered with final exams. For 45 years ND didn't participate in bowl games. In the '60 the polls started including the bowl games in the final poll which ranked the National Champions. Parshegian made the case to the Administration that ND had to compete in the bowl process or could go undefeated and be left out of the Championship rating after the bowl season. The goal was to face the highest ranked opponent.
Following the '69 season #9 ND went to the Cotton Bowl to face the#1 Longhorns. The next year ND lost the season closer to LSU and fell out of the Top 10. Despite a 8-2 season they chose not to go to any bowl that year.
In Devine's first year ND went 8-3 falling out of the Top 10 in the next to last game of the season to PIT and stayed home at bowl time. In '79 ND lost 2 of the last 3 including a rout by unranked Tennessee and again ND stayed home despite a 7-4 record.
ND was 8-3 in '96 but Holtz had resigned.
All those seasons were good enough to go to some bowl. Initially ND only considered the major bowls and a chance to move up in the ranking through playing a higher ranked opponent. In '83 an unranked 6-5 ND team went to the Liberty Bowl and knocked off #13 BC. In '84 the 17th ranked Irish (7-4) lost to #10 SMU in the Aloha Bowl.
In Davie's first year '97 an unranked 7-5 ND got beaten by 15th ranked LSU avenging a regular ND win.
By then the conferences had contracts with many of the bowls and ND's options were severely restricted. Teams playing in bowls also got the advantage of additional practice sessions in December. Bowl games also gave teams more media time which kept them in the front of recruits.