Are some Integer programming formulations completely useless for relaxation?
Yes, some IP formulations are less useful than others. The technique used to show that an LP relaxation can only be so good is showing integrality gaps. For a minimization problem, an integrality gap of $k$ in an instance in which the optimal integer solution has value $V$ but the LP has a solution of value at most $V/k$. This shows that any rounding mechanism cannot guarantee an approximation ratio better than $k$.

As a simple example, consider the relaxation of the vertex cover IP $x_i + x_j \geq 1$ for every edge $(i,j) \in E$ (the variables are $x_i$ for every vertex $i$, and they are constrained to lie in $[0,1]$). A triangle has minimum vertex cover 2, but the LP has a solution with value only 3/2 (namely, all vertices get the value 1/2). This is an integrality gap of 4/3. There are more complicated integrality gaps of $2-\epsilon$ for every $\epsilon &gt; 0$.