Yes, incorrect settings can break a monitor. Most significantly, refresh rate can cause serious problems. An incorrect resolution (number of horizontal pixels x number of vertical pixels) may be less likely to cause damage. But at least some monitors really dislike bad refresh rates.
Now, how likely is damage to occur? That can vary based on factors like how good of quality the monitor is. Ideally, if a monitor doesn't support certain settings, then it will just show a message saying "No signal", or something benign like that. What will break some monitors might not break other monitors.
I'm basing this on third party information that I've heard or read multiple times over many years. I have experienced some CRT monitors make interesting noises, like the snap of a spark. I was sane enough to not tempt fate and try to create a fire hazard, so I would immediately do something (probably pressing Esc on the keyboard, to undo the change made in a GUI... if that didn't have the desired effect, I probably turned off the monitor to try to prevent permanent damage.)
There may be a sticker on the back of the PC, near where the power cord goes, which may provide a list of officially acceptable refresh rates. Deviating from that list is not recommended. Often, newer video drivers try to only provide options specified by the monitor's driver. When I first got into computing, there usually weren't specific drivers for monitors, so such protections were historically less common.