To say "50MHz oscilloscope" is to say that it'll show signals accurately up to that frequency, but beyond that signals will be attenuated or not visible. This is a soft transition. No scope's electronics is perfect.
As with audio amps, the upper frequency limit is often defined where a sine wave input diminishes by 3dB, that is, loses half its power, compared to lower frequencies, when going from input to display. Read the scope's specs carefully to be sure.
Note that for complex signals or signals with sharp edges such as sawtooth waves, square waves, I2C signals, bluetooth signals, whatever, the upper frequency limit applies to the Fourier spectrum of the signal. Sharp transitions get mushy. The higher the upper frequency limit, the less mushy. A square wave right at 50MHz may appear so rounded off, but probably not quite as smooth as a sine wave.
For sampling rate, if the scope's amplifier and display are good up to 50MHz, and it's not an analog scope (dusty, old, vacuum tubes, ah good ol' days), the signal needs to be sampled at something like 4x or 5x that frequency to actually show a sine (or sine-ish) wave.
Nyquist says something about 2x, but think about it - if you sample a fast sine wave at 2x its frequency, you could by chance be sampling it once as it crosses zero on the way up, and again as it crosses zero going down. Then you'd see flat zero. So sampling is more than 2x. Nyquist still applies, but in having the response of the scope drop greatly before half the sample frequency. For your 50MHz / 250MHz example, there's probably a fast drop in response somewhere between 100MHz and 120MHz. Beyond that range, you won't see anything.