We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameter, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximityoblivious testers. While proximity-oblivious testers were studied before - most notably in the algebraic setting - the current study seems to be the first one to focus on graph properties. We provide a mix of positive and negative results, and in particular characterizations of the graph properties that have constant-query proximity-oblivious testers in the two standard models (i.e., the adjacency matrix and the boundeddegree models). Furthermore, we show that constant-query proximity-oblivious testers do not exist for many easily testable properties, and that even when proximity-oblivious testers exist, repeating them does not necessarily yield the best standard testers for the corresponding property.