In this work we prove non-trivial impossibility results for perhaps the simplest non-linear estimation problem, that of {it Group Testing} (GT), via the recently developed Madiman-Tetali inequalities~cite{madiman-tetali}. Group Testing concerns itself with identifying a hidden set of $d$ {defective} items from a set of $n$ items via $t$ {disjunctive/pooled} measurements (``group tests”). We consider the { linear sparsity regime}, i.e. $d = delta n$  for any constant $delta >0$, a hitherto little-explored (though natural) regime. In a standard information-theoretic setting, where the tests are required to be { non-adaptive} and a small probability of reconstruction error is allowed, our lower bounds on $t$ are the {it first} that improve over the classical counting lower bound, $t/n ≥ H(delta)$. As corollaries of our result, we show that (i) for $delta approxgeq 0.392$, individual testing is essentially optimal, i.e., $t geq n(1—o(1))$; (ii) there is an { adaptivity gap}, since for a range of $delta$  known {adaptive} GT algorithms require fewer tests than our lower bounds for non-adaptive GT; and (iii) similar lower bounds can be derived for the related partial recovery problem. Perhaps most importantly, our work provides a framework for combining combinatorial and information-theoretic methods for deriving non-trivial lower bounds for a variety of non-linear estimation problems.