The R g -conditional diagnosability of a multiprocessor system modeled by a graph G, denoted by t R g ( G ) , is a generalization of conditional diagnosability, which restricts every vertex contains at least g fault-free neighbors. Particularly, the R 1 -conditional diagnosability is the conditional diagnosability. The R g -conditional connectivity of a graph G, denoted by κ R g ( G ) , is the minimum number of vertices, whose deletion will disconnect the graph and every vertex of G has at least g neighbors in the remaining subgraphs. In this paper, the relationships between the R g -conditional connectivity of a graph G and its R g -conditional diagnosability under the PMC and MM* models are explored. We establish the R g -conditional diagnosability t R g ( G ) equals κ R 2 g + 1 ( G ) + g under some reasonable conditions, except the R 1 -conditional diagnosability of G under the MM* model. Moreover, we show under the MM* model, t R 1 ( G ) = κ R 2 ( G ) with similar conditions. Applying our results, the R g -conditional diagnosability of the ( n , k ) -star graphs and the ( n , k ) -bubble-sort graphs are determined.