I would be more inclined to say that one of the cards in your SLI setup is self terminating the Nvidia video driver in some way MrJoshpillow.
And perhaps the cause of the message that the video card, or cards have been removed as far as BF 1 Is concerned
When running.
It would make sense if you have experienced any frame rate loss while playing the game.
As of course the cards will be stacked quiet close together in the SLI configuration of your tower based system.
As a simple test, if the side panel of the system case is attached, I would first remove it.
Then try running BF1 to see if the error happens again in any form or way you previously received.
If the system runs fine, and both of the GTX 770 cards in the SLI setup have been in the system for a good few months, and both happen to be of the Nvidia reference design with a single blower type fan configuration.
Remove both of the shroud or cowlings that cover the cooling solution from the card and the blower type fan setup of each card.
You will most likely find between the blower type fan, and the very front edge of the graphics cards fin cooling solution a thick, or very clogged buildup of dust covering the cooling fins of the cooling solution on each card restricting air flow.
If the case it is likely that either one of the cards Gpu`s is getting very hot snd trying to down clock in speed in order to keep the gpu chip cool enough.
If it fails to do this as a last resort it will self terminate the video driver so it can get rid of the excessive heat building up on one of the Gpu chips in use.
As said leading to the low fps at first, then the error message from BF1 about the hardware being disconnected from your system ok.
By what you have described this is what can happen if the internals, or should I say the fan shroud of both cards or the cowling`s have never been removed for cleaning of both cards if in use if used for along time in months, or years.
If you look for what I have described as the possible cause you will see how bad the blocking of the front end of the cooling solution of the card is effected just after the single paddle wheel fan design of the reference built cooling solutions of the GTX 770 cards can get.