Is it guaranteed that there will be some $p$ such that $p\mid2^n-1$ but $p\nmid 2^m-1$ for any $m<n$?
In other words, does each $2^x-1$ introduce a new prime factor?
Yes, it's true (except for $2^6-1=7\times 9$).
This is known as Bang's theorem, and is a corollary of Zsigmondy's Theorem.
You can find a proof here (Theorem 3).
This result is due to Zsigmondy (1892), with special cases (b=1) (re)discovered by Bang (1886), Birkhoff & Vandiver (1904) ... see Ribenboim, The New Book of Prime Number Records, p. 43, 67-68, 338, 437. Vandiver, a prolific researcher on FLT, applied it to Fermat's Last Theorem (first case) and related diophantine equations, e.g. see Ribenboim's book 13 Lectures on FLT, pages 52,161,206,234,236.
Such results have many applications: a MathSciNet search on Anywhere=(Zsigmondy or (Birkhoff and Vandiver)) will find over 35 related Math Reviews. Schinzel (1974, MR 93k:11107) extended the theorem to arbitrary algebraic number fields $K$:
Later (1993, same MR) "he generalized this to show that every algebraic number which is not a root of unity satisfies only a finite number of independent generalized cyclotomic equations considered by the reviewer [in Structural properties of polylogarithms, Chapter 11, see p. 236, Amer. Math. Soc., Providence, RI, 1991; see MR 93b:11158]".