Adams had last been on the losing side when Portsmouth defeated former club Northampton on 19 December last year.
His unbeaten run was the longest in the Football League since Scunthorpe's Eddie Nolan went 33 games in 2013-14.
Arsenal's Sol Campbell holds the record in the Premier League, playing 56 games between defeats.
"It's mad that I haven't lost a game of football for so long," 30-year-old Adams said earlier in the week. "The lads keep asking me, 'How many is it now?'
He played 17 league games last season after Northampton bounced back from their loss against Pompey to win promotion to League One.
This season he had made 15 appearances for Carlisle before Saturday's game, scoring once.
Boss Keith Curle predicted that his side would be "dangerous animal" after their first league reverse of the season.
He told BBC Cumbria: "We've had a phenomenal start to the season - 16 games, undefeated in 16. Now we start again."
Carlisle United midfielder Nicky Adams tasted defeat for the first time in 33 league games as they were beaten at Newport County on Saturday.