Let \(C\) be a smooth irreducible complex projective curve of genus \(g\ge 2\). The Clifford index of a line bundle \(A\) on \(C\) is the integer \(\mathrm{Cliff}(A)=\deg (A)-2\cdot (h^0(A)-1)\). The Clifford index of \(C\) itself is defined as \(\mathrm{Cliff}(C)=\min \{\mathrm{Cliff}(A)\mid h^0(A)\ge 2, h^1(A)\ge 2\}\). Clifford's theorem says that \(\mathrm{Cliff}(C)\ge 0\); on the other hand \(\mathrm{Cliff}(C)\le [(g-1)/2]\), where \([\,]\) is the least integer function, equality holding for a general curve of genus \(g\). The main result of the paper states that if \(C\) lies on a complex projective K3 surface, then \(\mathrm{Cliff}(C') = \mathrm{Cliff}(C)\) for every smooth \(C'\) in the linear system \(| C|\). Moreover, if \(\mathrm{Cliff}(C)< [(g-1)/2]\), then there is a line bundle \(L\) on \(X\) such that \(\mathrm{Cliff}(L_{C'}) = \mathrm{Cliff}(C')\) for every smooth \(C'\in | C|\). The method of proof relies on a vector bundle construction due to the second author [J. Differ. Geom. 23, 299--307 (1986; Zbl 0608.14026)] and on a Clifford-type inequality the authors establish for vector bundles on a regular surface.