...Okay.
CRTs aren't instantaneous. They're constrained by their refresh rate and their raster scan method.
At 60 Hz, a new frame gets sent from the source every 17ms, and then that frame is drawn to the screen line by line, pixel by pixel:
You can take the same lag testing equipment that modern websites use to test LCDs and, through some converters, use them on CRTs as well:
The tester interacts with the display in two ways:
1) It sends a test pattern to the screen via its HDMI port. (If the screen doesn't have an HDMI input, then you have to chain it through a converter of some kind.)
2) It has a photo-sensor that you place over each of the three white bars displayed on the pattern (top, middle, and bottom). This measures the total time elapsed from (a) when the tester fired off the test pattern over HDMI and (b) when that test pattern is finally drawn on the screen.
Because CRTs (and most flatscreens, actually) draw their picture line by line, you'll observe faster readings on the top and slower readings on the bottom. Because of the 60 Hz refresh rate, the difference between the start and end point of the frame is close to 17ms. The difference between the top and bottom bars of the above snapshot is roughly 14ms, which sounds about right when you take into account overscan and other minutiae. Average the above values together and you'll get
7.9ms, which is the total lag rating that this screen would receive from a website like Displaylag, not 0ms.
And that's the floor for a CRT. In theory, it's possible for future display tech to drop to lower values than this if the refresh rate increases beyond 60 Hz (not only for TVs, but for the consoles and other source devices that feed them the picture in the first place). Can't say that I ever expect it to happen though.