We present a new property called constraint looseness and show how it can be used to estimate the level of local consistency of a binary constraint network. Specifically, we present a relationship between the looseness of the constraints, the size of the domains, and the inherent level of local consistency of a constraint network. The results we present are useful in two ways. First, a common method for finding solutions to a constraint network is to first preprocess the network by enforcing local consistency conditions, and then perform a backtracking search. Here, our results can be used in deciding which low-order local consistency techniques will not change a given constraint network and thus are not useful for preprocessing the network. Second, much previous work has identified conditions for when a certain level of local consistency is sufficient to guarantee a network is backtrack-free. Here, our results can be used in deciding which local consistency conditions, if any, still need to be enforced to achieve the specified level of local consistency. As well, we use the looseness property to develop an algorithm that can sometimes find an ordering of the variables such that a network is backtrack-free.