In this paper, we study the restricted connectivity and restricted fault-diameters of locally twisted cubes under the condition that each node has at least one fault-free neighbor. First, we prove that under the condition that if each node of an $n$-dimensional locally twisted cube $LTQ_n$ has at least one fault-free neighbor its restricted connectivity is $2n-2$, the same as that of the $n$-dimensional hypercube. Then, we give an upper bound on the restricted fault-diameter of $LTQ_n$, that is, the restricted fault-diameter of $LTQ_n$ is no more than the fault-diameter of $LTQ_n$ plus 6.