In NFL history, the most consecutive losses is 26 by the Tampa Bay Buccaneers in 1976 and 1977. Tampa Bay's first year in the NFL was 1976 and they lost all 14 games on the schedule. The following year they lost the first 12 games on the schedule before finally defeating the New Orleans Saints 33-14 for their first NFL victory.
26 consecutive losses by Tampa Bay from 1976-77 http://www.football.com/nfl/records/NFLTGamesLost.shtml
The Indianapolis Colts
Tampa Bay Buccaneers - 13 in the 1976 and 1977 seasons.
Tampa Bay Buccaneers with 26 consecutive losses in 1976-1977.