It depends on what your game actually is. It is obviously better to use smaller sprites and repeat them, but sometimes this is just not possible, for example if you have a baseball game and you need to see most of the stadium. You basically have no choice but to have one big stadium background, and then layer smaller sprites on top of that.
I try to keep these down to 1920 x 1080 and it works fine.
If you need larger backgrounds you may want to try a single 1920 x 1080, then place 2 instances in your layout, next to each other, and flip one horizontally making sure there is no seam where they meet ( this works well with nature scenery, but may not work with other stuff ). This would give you a background of 3840 x 1080 while still using a single 1920 x 1080 image
Then you can add other sprites on top of that to give it some variation.
But, like I said it all depends on what the game is. There may be a better way, there may be no way around it.