When does server-side rendering, or SSR, actually slow things down instead of speeding them up?

We're talking about server-side rendering for a new app people will use. Everyone pretty much agrees it makes pages load faster at first and helps with search engines, which is awesome for our public stuff. But, I'm kinda worried about how much it might slow down our server, especially when more users come online and pages change a lot. So, like, when does rendering on the server just become too much trouble and not worth the speed boost? Are there certain apps or times when a client-side approach, maybe with some pre-rendering for the unchanging parts, just works better for performance in the long run? I'd love to hear from anyone who has actually scaled applications using SSR.