As Chrispy mentions consider Jupiter. I believe it would not have needed to be much bigger than it is now to have "fired up".
However if you are dealing with the two stars having an elliptic orbit about their common centre of gravity then that is not quite a parallel to a a solor system where Jupiter is massive enough to be a binary star. Planets orbiting one star would travel round the elliptic with that star. I think the main issue is just how close they come to the centre of gravity and whether that puts the one star within the diameter of the orbits of the other star's planets:
Imagine a bunch of planets orbiting just one of the stars in the picture above (from wiki - see my earlier post for the link). If the two stars come too close then the planetary orbits could be disrupted. You could even have a situation where one sun "steals" a planet from the other sun as it passes. I guess a lot would depend on how long they remain that close and where the planet in question is at their closest approach. This is why I tend to think it would only be stable if the two stars' separation is quite large and their period of orbit also quite large. This would make it more likely that they never come that close. Though by no means certain, it would depend on just how elliptic the orbits are and how "tight" the overlap of those orbits is.
However if you are dealing with the two stars having an elliptic orbit about their common centre of gravity then that is not quite a parallel to a a solor system where Jupiter is massive enough to be a binary star. Planets orbiting one star would travel round the elliptic with that star. I think the main issue is just how close they come to the centre of gravity and whether that puts the one star within the diameter of the orbits of the other star's planets:
Imagine a bunch of planets orbiting just one of the stars in the picture above (from wiki - see my earlier post for the link). If the two stars come too close then the planetary orbits could be disrupted. You could even have a situation where one sun "steals" a planet from the other sun as it passes. I guess a lot would depend on how long they remain that close and where the planet in question is at their closest approach. This is why I tend to think it would only be stable if the two stars' separation is quite large and their period of orbit also quite large. This would make it more likely that they never come that close. Though by no means certain, it would depend on just how elliptic the orbits are and how "tight" the overlap of those orbits is.