I know that with a neutral nucleophile on a second or third leaving group you get a mixture of both Sn1 and E1. So does that mean you never have only an Sn1 reaction without also having E1?
SN1 and E1 often occur together because the nucleophile/base is weak, but we can control our conditions to favor either SN1 or E1. Adding heat to a reaction will favor elimination. Solvents also have an impact on what kind of reaction will take place.
Hope this helped!