In 2011, Chen computed the greatest common divisors of consecutive shifted Fibonacci numbers F_n + a and F_n+1 + a for a ∈ {±1, ±2}. He also showed that gcd(F_n + a, F_n+1 + a) is bounded if a ≠ ±1. This was later generalized by Spilker, who also showed that gcd(F_n + a, F_n+1 + a) is periodic if a ≠ ±1. In this article, we compute the greatest common divisor for a = ±3 and we show how the results given in this article compare to bounds derived by Chen and periods derived by Spilker. We further give a necessary criterion for an integer d to occur as such a greatest common divisor.