The term 'jump the shark' has become a little cliche, but it exists for a reason. Sometimes, continuing a story is a bad idea. Once you've milked the concept for all the quality it provides, and dealt with all the situations the story demands, you're probably well off ending it and starting on something new.
*To those of you unfamiliar with the term, 'jump the shark' refers to when most people say the TV show Happy Days started to suck. Fonzie was in a surfing competition, and a shark attacked him, and he used his surfboard to jump over the shark. It was supposed to, like, funny, or cool or something, but it wasn't. The show thereafter died whimpering.
I think it depends on the structure of the game. To continue the TV example, some games are like Star Trek, some are like the Twilight Zone, and some are like Babylon 5.
Star Trek-ish shows have a loose concept, and storylines tend to be self-contained. There is little in the way of a story 'arc' in the game. You can keep on going with the game as long as you can keep on coming up with new ideas for adventures that let the characters show off. Indeed, the primary aspect of the game is the characters -- how they interact, what their personalities are, how they deal with adventure. While characters will grow and change a bit, there is no clear-cut goal in the game. It is possible, however, to write yourself into a corner, and set the cast of characters down a plotline that just ruins the point of the game. Like, imagine if, for some sort of drama, they'd decided that Captain Picard died, and never came back. It might make for a good story for a while as the ship copes, but you'd really have ruined a lot of what made the show unique.
Similarly, for a Star Trek-ish show, the best reason to end is that the cast changes too much. If too many PCs die, or if too many new characters join too quickly, the feel of the game changes. You get a sort of psychic conflict between old plotlines and new characters, since almost inevitably the two won't mesh. This happened in my latest game, where we lost three of the four initial PCs over the span of eight sessions. While we also replaced them with new characters, for a while there was a strain to make the one remaining original PC's motivations fit with the motivations of the new characters.
An important thing for keeping a game like this going is that, if characters do change, that you encourage the players to create characters that have a reason to continue with the group. Unless you really don't care about thematic consistency in the story, don't let them just make up new characters with no connection.
Twilight Zone-ish shows thrive on quick change. You might have one or two plot threads repeat themselves over the course of several years of gaming, but most adventures are self-contained, and even the characters involved are not as important. Indeed, compared to Star Trek-ish games, Twilight Zone-ish games are much more story driven, and less character-driven. These types of games can easily make use of published adventures, because those adventures are cool, new, and original. The goal is to entertain week after week, and the GM for this sort of game doesn't worry much about telling a single long story, but rather many short ones.
It's really hard to 'jump the shark' in this sort of game. While individual adventures might not be as good as the rest, it's easy enough to get a new start with the next adventure. Players can come and go, settings change, and even GMs can switch sometimes. The goal for the group is just to game and have fun, and variety is the spice of life.
Babylon 5-ish games tie character and story together intricately. There is one story being told, and though it might have many twists and turns along the way, everyone has a sense that eventually there will be an ending. Having a character leave can be disastrous, requiring whole plot threads to change. GMs of this sort of game are often the most adverse to PC death, not just for the sake of the group, but for the sake of their own story.
Babylon 5, in my opinion, jumped the shark at the beginning of the 5th season. It was obvious the production crew changed, what with the production values seeming so much different between 4th and 5th season. Two characters left the show, one through death, one through contractual issues. Heck, it's pretty common knowledge that J. Michael Straczynski rushed season 4 because he wasn't sure he'd get a 5th season, and thus most of the plot came to an end at the end of season 4. The point of the show had been fulfilled, and continuing it, though it did indeed lead to some entertaining moments, sadly did not keep up the same feel of the show from before.
This sort of game is not the type of thing you want to run while in college, trust me. They work best either with a small, dependable group (two or three players), or when all of your gamers are settled, have families, and aren't likely to skedaddle off to California.
I suppose we should have a more techinical term. Let's just say we want to measure a campaign on several factors, and one of those factors is the Arc, or lack thereof. On a scale of 1 to 10:
Star Trek: The Original Series - 4
Star Trek: The Next Generation - 5
Star Trek: Deep Space Nine - 6
Twilight Zone - 1
Babylon 5 - 9
Angel - 7
Sagiro's Storyhour - 8
Piratecat's Storyhour - 5
(Contact)'s Return to the Temple of Elemental Evil Storyhour - an odd mixture of 2 and 8. 2 for the character turnover, 9 for the overarcing plot. Once they defeated the temple, there wasn't really much of a reason to keep going. But, oddly, I hear they did. I never really looked into it. Hmm.
