If they existed at the same time then wouldn't there be some evidence of that? What form would that evidence take? Maybe some bones of people and dinosaurs buried at the same level underground? If they existed at the same time there would be some way to prove it. It's up to the people who are saying this to prove it because it is impossible to prove a negative. If I say that humans and dinosaurs didn't exist at the same time there's no way I can prove that because you can't prove a negative. I'm trying to think of a way to illustrate this. Let's say you think someone broke into your house. If something was taken you have proof that they WERE in your house. But you can't prove that they weren't. You can say, "I see no evidence that anyone was here" but just because there is no evidence doesn't mean they weren't. You can't be scientific about it because there is no way to test that theory. It's just the nature of the question one is asking.
As a Creationist and a Christian It is a particular hobby of mine learning how to defend my beliefs in a scientific way. A lot of the evidence is taking the same things that people use to prove evolution. (I'm talking marco evolution not Micro. Micro Evolution can be proven and is a fact. Macro however is a theory and a belief.) As a Creationist I start with the belief that God created the earth in 7 literal days. Evolution starts out with the belief that the earth is millions of years old. Both sides start out with a belief and then use the evidence to back up that belief.
The simple answer is that humans and dinosaurs lived at the exact same time. They were actually created on the exact same day.
I will happy to answer any questions any one has about what I believe if they are asked in nice way.

