Generally no. Assuming a typically setup, you are most likely watching or listening to only one thing. For example, I have Steam and hence my games on my 1TB SSD. I have music and what not on an 8TB spinner. If I'm playing a game, I load up an album and have it play. I'm listening in real time while playing my game, how fast do you think that drive needs to be? It's doing one thing, and if it did it too fast I wouldn't want to hear it. Fun fact, DVD players back in the IDE days only bothered with ATA 33. Because that's all the bandwidth a DVD needed. Assuming your spinner is a modern drive attached to a SATA III port, or 600MBps. Trust me, your data files are fine on a spinner.
This answer changes if you are not talking about a normal setup. If your OS, Game, and music all sit on a spinner and you listen to music or have a movie player on the TV to entertain the kids I can see issues there with game smoothness. Same if this is some sort of home server with everyone trying to stream your itunes account. But if it's just you just doing one thing I'm sure you are fine.