Let's say you and your family had a secret. You could read each other's minds during times of high stress. You can't really prove it to people and during the times of high stress, no one would be able to be calm enough to prove it.
Now, your sister hit a tree in a rural area with her car and her child was in critical condition. She didn't have a cell phone, but there was a store 100 or so feet away she could run to to call 9-1-1. Instead of doing so, she spoke to your other family members in her mind and your brother had a cell phone. He called 9-1-1, but accidentally told them the wrong street. When they didn't show up and your sister thought it was taking too long, she ran to the store and called and by the time they got there and got your sister's child to the hospital, it was too late.
People you know who heard about it, including your wife, think that your sister just sat there doing nothing and took way too long to get to the store to the phone. They, understandably, think of her as neglectful and didn't do all she could have to save her child. They don't want to tell this to her face, though, but don't want to be around her. They of course wouldn't believe the outrageous claim that we can speak to each other with our minds. What does one do?
(obviously an analogy. we can't really speak telepathically during high stress or not and nothing happened to my sister and she doesn't have a child)