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What make/model of set is it?
And I say again, check your local oscillator with another radio. It is not likely, but it may be stopping at the low end of the band. Getting noise, however, makes this unlikely, but I am a b it nutty about this because if I had done it once, I could have saved myself a days work trying to find a RF amp problem, when I actually had an oddball intermittent oscillator.
Does the reception get worse the closer that you get to the low end of the dial, or does it abruptly stop at one point?
Sudden stopping at one point on the dial is probably caused by the tuning cap plates shorting out, due to mechanical damage (a bent or nicked plate) or metal trash (fragments from the rotor shaft ball bearings, etc.) lodging between the plates.
A gradual loss of signal as you move to the low end, however, could be caused by an inadequate antenna. If the set aligns OK with a sig gen at 600 Khz, but still won't pick up over-the-air signals at the low end, my guess would be that you need a better antenna, or something about the building that you are in is attentuating the longer-wavelength signals (for instance, a metal roof, metal siding, or steel-beam construction).