Biggest thing that can happen: You may not have gotten a complete list of suspects.
The first thing I have to ask is for people to finger someone else. Even just today, I had one where there were already 5 suspects + my client. It was a Really Hard one, which means I was missing people. I pressed my one question on the last person on my list to get more names and Boom! She coughed-up 2 more suspects! She was my only link to them and guess what!?! She was killed off later in the case.
If you do not know all of the suspects, the killer's name may not be on your list. Thus, you cannot possibly find them. :-(
Lock Pick and Research save you questions. I'm sure others can also suggest more and probably better tactics.