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Jelly, 15 bytes

e€ØDċⱮØ.ẇ⁸LŻÆḞ¤

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How it works

e€ØDċⱮØ.ẇ⁸LŻÆḞ¤ - Main link. Takes a string S on the left
  ØD            - Yield "0123456789"
 €              - For each character in S:
e               -   Is it a digit?
                  Call this new list of 0s and 1s B
      Ø.        - Yield [0, 1]
     Ɱ          - For each in [0, 1]:
    ċ           -   Count in B
                  This yields a pair [a, b]: [# of non-numeric, # of numeric]
              ¤ - Create a nilad:
         ⁸      -   Yield S
          L     -   Take its length, L
           Ż    -   [0, 1, 2, ..., L]
            ÆḞ  -   nth Fibonacci number for each
        ẇ       - Sublist exists?
                  This returns 1 if the left argument ([a, b]) is a contiguous sublist
                  of the right argument (the list of Fib numbers), i.e.
                  a and b are both Fibonacci numbers, and a immediately precedes b