It is possible for person with AIDS be negative for HIV antibodies?
Usually an antibody test against HIV is positive three month after the infection. Before this time a test can only be done with molecular biology methods as PCR. However, it is possible, that patients which have definitely been tested in an early phase using molecular biology methods and which afterwards receive a antiretroviral therapy (ART) that these patients do not develop high serum levels of antibodies, as the virus titer in the blood is not high enough for a sufficient reaction and thus stay undetected by the tests. This has been improved, so that the kits are supposed to have a higher sensitivity now. See these papers for more details:


Seroreversion in subjects receiving antiretroviral therapy during
acute/early HIV infection.
The effects of early antiretroviral therapy and its discontinuation
on the HIV-specific antibody response.


I haven't found any papers on the decline of HIV-antibodies in AIDS (which is the late stage of the disease), but if they do, this is probably due to the effects of the infection on the immune system.

Low counts in CD4 positive T cells (which are usually associated with an HIV infection) can also occur with other diseases. See these papers (especially the first, as this is a review which contains a lot of further references):


Idiopathic CD4 Lymphocytopenia: Spectrum of opportunistic infections,
malignancies, and autoimmune diseases.
Decreased CD4 and increased CD8 counts with T cell activation is
associated with chronic helminth infection.
Acquired immunodeficiency without evidence of HIV infection: national
retrospective survey.