Given a word rewriting system with variables $R$ and a word with variables $w$ the question we are interested in is whether all the instances of $w$ obtained by substituting its variables by non-empty words are reducible by $R$. This property is known as {\em ground reducibility} and is the core of the {\em inductive completion} methods that have been designed for proving theorems in the initial model of equational specifications. We prove the problem to be generally undecidable even for a very simple word $w$, namely $axa$ where $a$ is a letter and $x$ a variable. When $R$ is left-linear, the question is decidable for arbitrary (linear or non-linear) $w$.