Summary: As we have announced in the title of this work, we show that a broad class of evolution equations are approximately controllable but never exactly controllable. This class is represented by the following infinite-dimensional time-varying control system: $$ z'=A(t)z+B(t)u(t), $$ $z(T)\in Z$, $u(t)\in U$, $t>0$, where $Z,U$ are infinite-dimensional Banach spaces, $U$ is reflexive, $u\in L^P([0,t_1],U)$, $t_1>0$, $1<p<\infty$, $B\in L^\infty([0,t_1],L(U,Z))$ and $A(t)$ generates a strongly continuous evolution operator $U(t,s)$. according to {\it A. Pazy} [Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. New York etc.: Springer-Verlag (1983;

Zbl 0516.47023)]. Specifically, we prove the following statement: If $U(t,s)$ is compact for $0\le s<t\le t_1$, then the system can never be exactly controllable on $[0,t_1]$. This class is so large that includes diffusion equations, damped flexible beam equation, some thermoelastic equations, strongly damped wave equations, etc.