I've been playing at the Nation level for a while.  I generally get to #1 in fame and can hold it until the last 2 eras.  At that time, I generally choose science civ's like the Joseon, French, Swedes etc.  No matter how much I invest in science and money, one of the AI civs always surges ahead of me in almost all the FIMS.  What gives with that?  More importantly, how can I combat against it?

Can you share some end game graphs?

More cities usually does it for me

Number of cities doesn't seem to matter.  I'm always at one over cap and I very often use the Archemenedid (?) Persians to get the +2 cap.

I think this is a saved file that will illustrate.  If you immediately resign you can see the graphs.  I played the Chinese here because the leading civ declared war and I needed the money to upgrade my armies fast since they had gotten ahead of me in science and were killing me.

Chinese Turn 178.ctr

AI get bonus FIMS depending on number of unlocked techs and how bad their fame is. Basically more unlocked tech more bonus, while more fame will decrease this bonus. The increase is a percent bonus.

That must be a MAJOR bonus.  If you look at the game I posted, my nearest competitor's surge was phenomenal!  How could I possibly keep up with that?

Many of the late-game cultures DO give big bonuses, I think.