This is because of the view clipping distance. The default values of the clipping distance are set for a reasonable distance for most purposes, but for things that are too big, like mountains, you can set to whatever distance you wish:
Make sure you also change the clipping distance settings in your camera or else you might have trouble seeing the objects in the render:
Be aware that depending on the size of the range you set, it can get less precise and some z-fighting may happen (that's when the software gets confused as to what face is in front of the other face, causing visual glitches).*
Another option is to just not make the mountains real-life scale. Make the whole scene smaller and no one will notice.
*Just as a curiosity: This can happen because the bigger the numbers the computer is dealing with, the less precise it gets because the computer can only store a limited amount of digits for each value. More digits before the point means less digits after the point, that's why huge numbers can lead to precision errors. This is something all computers have to deal with, not something specific to blender.