First and most importantly, if you have not been assigned a project manager on a remodeling project yet, ask who it will be and if you will have his personal cell phone number.
It is imperative that any type of remodeling project have a person you can contact at ANY TIME for emergencies, as well as one that can be contacted during business hours for standard inquiries.
When you begin having a home remodeling project performed, your excitement and anticipation for the results will be high and it should stay that way. To have something such as waking up on a Saturday morning to discover you have no hot water in part of your house should not be ruining the experience.
Worse yet would be to not be able to reach a project manager if something did go wrong. Being human, things can and do happen. In the example above, suppose a worker forgot to turn on an important valve after performing a task. That problem should be corrected quickly...not on Monday.
Unfortunately there is no foolproof way to test the responsiveness of your project manager ahead of time. However, you can gain clues from your interaction with that person. If you are getting slow response from them before they even have your deposit, that's a good indication you should seek a different contractor.
During my larger projects inside someone's home, they usually have me entered on speed dial and can expect an immediate answer when possible, or, can expect a call or email response within a few hours to answer questions. A good project manager will “live the experience with you” and protect the reputation they work hard to achieve.