Clipping in the 3D Viewport:
It sounds like you have your camera's Start Clipping Distance set too high in the 3D Viewport Properties. Clipping distances define the range (in distance units) within which objects are visible. If your distances are set incorrectly, your view gets "clipped" when you cross the clipping threshold, as you describe. To change the 3D Viewport clipping settings:
Properties panel (N) -> View section -> set the Clip Start distance lower
(NOTE: These settings only effect display when NOT in Camera view)
You can read more about clipping in the 3D Viewport here.
Camera Clipping Settings
Just so you know, there is also the option to adjust the clipping range of a camera. This effects the scene when viewed through the camera, such as in Camera View or when rendering. To change these properties:
Select camera -> Object Properties window -> Camera tab -> Lens section -> adjust the Clipping Start and End values
You can read more about camera clipping distance here.