I would like to ask the following. Given only the leading terms of an ideal I, namely the set $LT(I),$ is it possible to find a Groebner Basis of I? If not always, then when is it possible? 
We know that $< LT(I) > = < LT(g_1),...,LT(g_n)> $ for a Grobner basis $g_1,...,g_n$ but can we find exactly one basis $ g_1,...,g_n $ given only the $LT(I)$ ?