It just depends on how the CRT is designed testproggie. However an ability to actually scan at different rates on the screen is one thing that distinguishes CRTs from so-called "Fixed Pixel Displays" which only have a single native resolution. IOW, CRTs do have the capacity to change the number of "scanlines" they display on the screen. FPDs don't, because they don't really use scanlines at all. They use pixels.
Most CRT computer monitors are what's known as "multiscan" or "multisync". If you've ever owned a CRT computer monitor, this is probably something you're familiar with. If you change the resolution in the display properties, the CRT computer monitor will actually adjust to a different scanrate and display a different number of lines on the screen. IOW they can display at multiple scanning rates.
All tubes have this capacity, but the other electronics that drive the tube place certain limitations on how many different rates it will support. In general, the larger (and flatter) the tube is, the more costly it is to implement a variety scanrates. It's also generally cheaper to produce large tubes that scan at lower rates than higher rates.
Your old analog TV for example only has the electronics for a single scanrate, namely 15.75khz, which gives you 480i. Which makes it very cost effective to produce.
Most current HDTV tubes can actually scan at two rates on the screen: 31.5khz (which gives you 480p and sometimes also 960i) and 33.75khz (1080i, and also 540p on some TVs). To enable the direct display of 720p on the screen, the tube must have the electronics to support a higher scanrate: 45khz. Upgrading the electronics to support a third and higher scanrate is expensive though. So it's usually cheaper for the manufacturer just to add a chip to digitally convert 45khz-720p to either 33.75khz-1080i or 31.5khz-480p for display.
(In Fixed Pixel Displays btw, all signals are digitally converted to the resolution of the screen, which is typically around 1280x768 pixels these days.)
Even though most tubes can support multiple scanrates, they will usually be designed in such a way that there's one optimum scanrate that looks best. The optimum scanrate depends on a variety of factors including the dot-pitch of the screen, beam spot size, decay rate of the phosphors, etc. In the case of HDTVs they may sort of split the difference so both 31.5khz-480p and 33.75khz-1080i look fairly decent. Or they may favor one or the other (hopefully the 1080i).