If you do a Google image search for the terms "Studio 60" and "The West Wing" together, you'll find that "Bartlet 4 America" poster in the backstage dressing room that was seen in this past Monday's episode of 'Studio 60'.
I was going to add 'Studio 60' to the shows in the dimension of 'The West Wing'. There aren't that many; the major ones are 'Mr. Sterling' and 'Smallville'.
There is a slight hitch, however, to that idea. The great outside world of 'Studio 60' seems more in synch with our world than that of 'The West Wing'.
For instance, Tom Jeter's younger brother is in the service and is serving over in Afghanistan. Over in 'The West Wing', the major conflict for American troops was in the buffer zone between China and Russia. Before that, Equatorial Khundu. But the Bartlet administration never invaded Afghanistan to throw the Taliban out of power; they never went after Saddam Hussein in Iraq - if he was even in power in that dimension.
There may have been a 9/11 attack in 'The West Wing' dimension, but it's never been brought to the forefront of their agenda. Details were hazy in that special episode "Isaac And Ishmael", so it could be it was nothing on the par of what happened to us.
So I'll go along with Will's suggestion that the "Bartlet 4 America" poster was a prop.... until something else comes along to toss 'Studio 60' out of Toobworld proper.
But it could also be the real thing - just not a campaign tool for Jed Bartlet of 'The West Wing'. It could be for any politician named Bartlet.
Maybe it's Dr. Josiah Bartlet who lived in Boston and was seen in at least one episode of 'St. Elsewhere'. It wouldn't be the first time a physician entered the world of politics.
Just ask Bill Frist.