Ok so you have a field called "tt" which is your travel times, lets assume they represent whole hours. When you are building your network you'll get to the stage of specifying your attributes, default is length. At this point you add a new a attribute, see image below.
Having added it right click on it and select Evaluators and set the type to field and value to your tt field. OK everything and network has your new travel time.
Use the network analyst toolbar to create a service area layer. Then in the properties change impedance to myTravelTime and set default breaks to 5, 10, 15, 20 hours (or what every you desire) as shown below.
Then add a location and solve and you get what is shown below.