While the barbed connector fittings are capable of 125 Psi water pressure, they are not very strong structurally, and excessive force when pushing the tube onto the barbs can easily break them.  They are not expensive, and we will gladly ship some more off to you in an envelope, but to push the tube onto the barb correctly there are two ways.
9 times out of 10 if someone complains about the tube coming off the barbed fittings  under pressure, it is either because the tube has been overheated (such as in a flame) before being pushed over the barb, or because the tube has not been pushed all the way over the barb.
In the pictures bellow, a good connection is contrasted against a poor connection.
The easiest way is to cut the tube off straight with sharp scissors, and to hold the tip of the tube in boiling hot water for about 10 to 20 seconds. The warmer the tube tip is, the easier it will be.  Boiling water does not damage or weaken the polyurethane.  Heat will cause the tip to be softened up, making it easy to slip over the barbs.  If for some reason you only get the tube over the barb partially before it cools down too much, and you can't pull it off anymore, you may need to consider redoing the joint.  This may take a little practice.  The polyurethane tube is flexible and stretchy, but this does underline the need to leave a little slack in your tube layout until you gain enough experience.
There is also a Pincher Tool you can make yourself.  This will allow you to push the tube over without any heat.  Check out Q18 for details.