Let us consider the quasilinear system \[ (1)\quad \sum^{N}_{i=1}\sum^{n}_{\alpha =1}D_{\beta}[A_{ij}^{\alpha \beta}(x,u)D_{\alpha}u^ i]=0,\quad in\quad \Omega \subset R^ n. \] It is well-known [see the first author, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Stud. 105 (1983; Zbl 0516.49003)] that the Legendre-Hadamard condition of ellipticity on (1) [\textit{C. B. Morrey jun.}, Multiple integrals in the calculus of variations (1966; Zbl 0142.387)] assures the validity of the Caccioppoli inequality [\textit{C. B. Morrey jun.}, J. Math. Mech. 17, 649- 670 (1968; Zbl 0175.119)] for the weak solutions of (1), in the case that the coefficients \(A_{ij}^{\alpha \beta}\) are sufficiently regular. However, this is not true any more in the general case. In fact, in the present paper the authors give two examples of linear (with bounded measurable coefficients) and quasilinear (with continuous coefficients) systems which satisfy the Legendre-Hadamard condition but whose solutions do not satisfy Caccioppoli's inequality.