Can one experience neutropenia / low blood counts from chemotherapy?

Yes. : Chemotherapy typically causes death of fast growing tissues. These include the cancer and a number of normal tissues including bone marrow, hair follicles, mouth cells. The damage to the bone marrow can cause low blood counts, especially neutropenia and low platelets.

Absolutely...: It depends upon the chemotherapy agent(s) being used. Many of these agents are myelosuppressive, meaning that they interfere with the bone marrow's ability to make new formed elements. Once counts fall to their nadir, they will recover as the stem/progenitor population produces new cells. Not all chemotherapy is myelosuppressive. Vincristine, for example, has little effect on blood counts.