This rule comes from UX design user studies. However, the actual rule is that people perceive an event happening within 100ms as "instantaneous". Or, in other words, two events happening within 100ms of each other won't feel like distinct events. This doesn't mean that users won't notice the delay and it doesn't even mean users won't be frustrated by it, it's just a matter of human perception of distinct events in time.

Unfortunately, programmers/UX designers have a tendency to generalize this rule and use it to excuse slow user interfaces.