With a little help from the trusty Baseball-Reference Play Index, I found every team in history to ever lose at least ten consecutive games more than once in the same season. It turns out, there have been 30 of them (this is for seasons going back to 1918, which is as far back as B-R's streak tracker goes).
Incredibly, the 1928 Phillies, 1961 Senators (they became the Rangers), 1962 Mets, and 1965 Mets managed to lose at least ten straight games THREE times during the season. Those 1962 Mets are especially (un)impressive, because their streaks were of 17, 11, and 13 games. That was the first season in franchise history for the Mets and they lost 120 games, the modern record.
That Senators team was also new to baseball when they had their streaks and it turns out that most of the teams that lost ten or more games in a row more than once did so before the expansion era.
In looking at the Astros recent (current) run of awful baseball, I discovered that the three worst 50-game stretches in modern history were all pulled off by the old Philadelphia Athletics. Those A's turn up twice among the 30 teams here, in 1937 and 1945. The Phillies are on the list for 1919, 1928, 1939, and 1961. The city of Philadelphia sure did see some losing baseball back in the day.
On the other hand, the Cardinals, Dodgers, Giants, Indians, Pirates, White Sox, and Yankees have all been around since before 1918 (which, as I said, is when this research dates back to) without ever having lost ten straight more than once in a season. That club may shrink from seven clubs to six this evening.
Back to the failures though, as I concluded in looking at the worst 50-game stretches in modern history, teams just aren't losing in bunches the way they used to. Only eight teams have accomplished this feat since the 1960s:
1989 Reds
1989 Tigers
1996 Tigers
1998 Marlins
2002 Orioles
2003 Tigers
2004 Diamondbacks
2006 Royals
Not a single team joined the list between 1970 and 1988, thought the late 90s and early 2000s saw a solid bump in membership. Those 1996 Tigers are one of only five teams in history to lose at least twelve straight games twice in one season, along with those 1962 Mets and 1928 Phillies, plus the 1919 Phillies and 1935 Braves. So, the '96 Tigers were the only team in the last fifty seasons to lose 12+ twice AND the only team in American League history ever to do it. The 1996 Detroit Tigers, ladies and gentlemen!
Can the Indians become the newest member of the club? I await with bated breath.