Resistance to punishment is commonly used to measure the difficulty in refraining from rewarding activities when negative consequences ensue, which is a hallmark of addictive behavior. We recently developed a progressive shock strength (PSS) procedure in which individual rats can titrate the amount of punishment that they are willing to tolerate to obtain food rewards. Here, we investigated the effects of a range of delays (0–12 s) on resistance to punishment measured by PSS break points. As expected from delay discounting principles, we found that delayed shock was less effective as a punisher, as revealed by higher PSS breakpoints. However, this discounting effect was not equally distributed in the population of rats, and the introduction of a delay highlighted the existence of two populations: rats that were sensitive to immediate punishment were also sensitive to delayed shock, whereas rats that were resistant to immediate punishment showed strong temporal discounting of delayed punishment. Importantly, shock-sensitive rats suppressed responding even in subsequent non-punishment sessions, and they differed from shock-resistant rats in anxiety-like behavior, but not in sensitivity to pain. These results show that manipulation of temporal contingencies of punishment in the PSS procedure provides a valuable tool to identify individuals with a double vulnerability to addiction: low sensitivity to aversion and excessive discounting of negative future consequences. Conversely, the shock-sensitive population may provide a model of humans who are vulnerable to opportunity loss due to excessive anxiety.