Much of sensitivity has to do with how well you know the person.
When we get uncomfortable or feel like we're being insensitive, it's normal to want to withdraw or fall silent or change the subject.
But sometimes, the behavior we think is insensitive isn't insensitive at all.
And our embarrassed reaction is the less-than-ideal behavior.
It's hard to always know because it varies from individual to individual, so it's impossible to get it right every time.
For example, a friend of mine told me of a blind man she knew (who'd fallen blind as a young child, so he knew what things looked like, colors, etc.). While interacting with him, people would forget he was blind and make comments like, "Oh, look at that! Isn't that just the most beautiful sunset?"
Then they'd realize their mistake and quickly change the subject.
But the blind man told my friend, "I know they mean well and all that, but every time something like this happens, I always feel like saying, 'Really? Could you describe it for me? Because I sure would like to see it, too!' "