Yes, that's precisely why you should never, ever, ever talk to the cops:
https://www.youtube.com/watch?v=d-7o9xYp7eE
I love that video. That first speaker really spells everything out in a concise and easily comprehensible way of how you can and will be fucked (if you are the prime suspect) if you open your mouth, innocent or not.
The first speaker is a lawyer. Then the second lawyer, a cop, says essentially "he's right. but you should trust our (cops') judgment, we are trying to help."
Right, you have to live with some level of trust. "Hey we are looking for a guy who kidnapped a child. Have you seen a white van in this lot today? It's really important we follow up on any possible leads so anything you can remember might help!"
"Sorry I won't speak to you without my lawyer present."
