Charcoal, 25 bytes

⊙⌕Ａθe⊙⌕Ａθn⁼✂θλ⊕ι÷⁻ιλ³nice

 ⌕Ａ                         Find all positions of
    e                       Literal string `e`
   θ                        In input string
⊙                           Any end position satisfies
      ⌕Ａ                    Find all positions of
         n                  Literal string `n`
        θ                   In input string
     ⊙                      Any start position satisfies
            θ               Input string
           ✂                Sliced in steps of
                  ι         End position
                 ⁻          Minus
                   λ        Start position
                ÷           Integer divided by
                    ³       Literal `3`
             λ              From start position
              ⊕ι            To end position "inclusive"
          ⁼                 Is equal to
                     nice   Literal string `nice`

The code tries to slice precinct from position 5 to position 2 but this doesn't work: "precinct"[5:2+1:-1] is just ni, therefore this slice is rejected as desired. Another rejection occurs when the start and end point are not an exact multiple of 3 apart, in which case the slice will undershoot the final e.

Charcoal, 25 16 bytes

⊙θ⊙θ№✂θκＬθ⊕μnice

 θ                  Input string
⊙                   Any index satisfies
   θ                Input string
  ⊙                 Any index satisfies
      θ             Input string
     ✂              Sliced in steps of
           μ        Inner index
          ⊕         Incremented
       κ            From outer index
        Ｌθ          To the end of the string
    №               Count i.e. contains substring
            nice    Literal string `nice`