There are certain stats indicative of a team's unsettled nature. That the Texans are the only team since their inception to have the first pick more than once is one. You might assume at first that this is another: Since the Texans' inception in 2002, they've drafted a lot of quarterbacks. The fifth most of any NFL team. They've taken six and four teams have taken seven.
But most of the time when you draft a quarterback, you aren't taking him to be a starter.
For Houston, it started with David Carr with the first overall pick in 2002. Then two quarterbacks in 2003: Dave Ragone out of Louisville in the third round and Drew Henson from Michigan in the sixth round. In 2004, the Texans took B.J. Symons from Texas Tech in the seventh round. Four years passed before the Texans took Washington State's Alex Brink in the seventh round in 2008. And the last quarterback the Texans drafted was T.J. Yates -- a fifth-round pick in 2011.
ESPN Stats & Information provided me this list of the teams that have drafted the most quarterbacks since 2002. Washington, San Francisco, Baltimore and Denver have taken seven in that span. The Patriots, Jets, Packers and Texans have taken six.
The Texans and 49ers are the only two that have taken one with the top overall pick/had a top overall pick in that span. The Texans and Jets are the only two with unsettled quarterback situations right now, though the Broncos and 49ers went through a lot of that during the past 12 years, and Washington isn't as secure at that position as the rest of the teams.