Generalization to AllTrue, AnyTrue and NoneTrue

Select is fairly close to this already, notably including early exit behavior, so perhaps:

someTrue[list_, pred_, {m_} | n_] :=
  n + m == Length @ Select[list, pred, 1 n + m]

Test:

someTrue[Range@10, PrimeQ, 3]
someTrue[Range@10, PrimeQ, 5]
someTrue[Range@10, PrimeQ, {4}]
someTrue[Range@10, PrimeQ, {2}]

True

False

True

False

---

The code above is me trying to be clever with vanishing patterns. The longer but more legible form:

someTrue[list_, pred_, n_]   := n == Length @ Select[list, pred, n]
someTrue[list_, pred_, {n_}] := n == Length @ Select[list, pred, n + 1]

Here's a way to implement this without Select, but with VectorQ. If you don't want to duplicate memory, then Select could be dangerous.

My idea was to count the number of times pred was true and exit once it hit's the specified threshold.

SomeTrue[list_, pred_, n_Integer] := Module[{i = 0},
  !VectorQ[list, (If[pred[#], i++]; i < n)&]
]

SomeTrue[list_, pred_, {n_Integer}] := Module[{i = 0},
  VectorQ[list, (If[pred[#], i++]; i < n+1)&] && i == n
]