Upto what DSLR camera model can I used the same lens?

I have a 55-200mm for my Nikon D3000 camera. If I buy a new camera, say D60 or D90, will I be able to use this lens on those camera without any distortions?

Added (1). Please also suggest a good camera… Not full frame.

I believe you can. On a side note, why would you want to go from D3000 to D60? If you want to upgrade you have better options.

The D90 would be a good camera to upgrade to, or the D7000. The D60 would be a downgrade, it is the long since discontinued predecessor of the D5000.

Yes, any lens that works on your D3000 will work on any other Nikon DSLR except for the full frame ones (technically they do work there too, but with serious drawbacks).It's actually the other way around, many lenses that work perfectly on other cameras such as the D90 or D7000 won't properly work on your D3000, such as lenses that take an internal autofocus motor in the camera body to autofocus. E. G. The 50mm/1.8D.
And the D7000 even can work with lenses that wouldn't even meter with either the D3000 or the D90… But for most users this is a non-issue, as these lenses are only available used nowadays, not new.

Before you upgrade though, ask yourself what you expect your new camera to do that your current one can't do. A new fine lens may do more for your photography than a new camera body.