Why some neurons are tetraploid
Neurons are tetraploid in pathological situations like Alzheimer disease:

Neurons that duplicate their DNA are rarely observed to undergo mitosis, and they remain for long time as tetraploid cells, in accordance with the chronic course of the disease. We have recently shown that cell cycle re-entry and somatic tetraploidization occurs during normal development in a subpopulation of RGCs (retinal ganglion cells), giving rise to enlarged neurons with extensive dendritic trees. Tetraploization in these neurons occurs in response to the activation of the neurotrophin receptor p75NTR by an endogenous source of NGF (nerve growth factor).

Source: Frade JM, López-Sánchez N. A novel hypothesis for Alzheimer disease based on neuronal tetraploidy induced by p75NTR. Cell Cycle 2010; 9:1934 - 1941; PMID: 20436277; http://dx.doi.org/10.4161/cc.9.10.11582

These neurons are generated in response to nerve growth factor (NGF) acting through the neurotrophin receptor p75 (p75NTR), which induces E2F1 activity and cell cycle re-entry in migrating RGC neuroblasts lacking retinoblastoma (Rb) protein. We have also showed that brain-derived neurotrophic factor (BDNF) prevents G2/M transition in the tetraploid RGCs, thus being crucial for the maintenance of the tetraploid status as well as the survival of these neurons. The realization that tetraploid neurons can be readily observed in the vertebrate nervous system has important physiological consequences.
Several eukaryotes are known to undergo endoreduplicative cycles leading to somatic polyploidy, thus increasing cell size in specific tissues. Examples of polyploidy in neurons are known in some invertebrates.

Source: José María Frade. Somatic tetraploidy in vertebrate neurons - Implications in physiology and pathology. Commun Integr Biol. 2010 Mar-Apr; 3(2): 201–203. PMCID: PMC2889987