Does neopentyl halide undergo SN1?
Neopentyl halides are not good electrophiles in the way that other primary alkyl halides are, primarily due to steric effects but also because of rearrangents and eliminations. 

This paper 1 notes in its abstract that "As a rule, the SN2 reactions of neopentyl bromide and tosylate with sodium alkoxides are extremely retarded by steric hindrance, and the yields of alkyl neopentyl ethers are poor because of side reactions such as neopentyl rearrangement, elimination, and decomposition." This is supportive of the view that an SN1/E1 process may be occurring. 

This paper 2 confirms rearrangements occur.

I don't think this leads to us being able to say neopentyl bromide undergoes SN1 reactions directly or easily. It is unreactive to nucleophiles; when it does react it does so through mulitple pathways leading to low yields.