In stream ciphers, we should use a t-resilient Boolean function f(X) with large nonlinearity to resist fast correlation attacks and linear attacks. Further, in order to be secure against an extension of linear attacks, we wish to find a t-resilient function f(X) which has a large distance even from low degree Boolean functions. From this point of view, we define a new covering radius p(t, r, n) as the maximum distance between a t-resilient function f(X) and the r-th order Reed-Muller code RM(r, n). We next derive its lower and upper bounds. Finally, we present a table of numerical bounds for p(t, r, n).