That is why you can fill a glass with water past the rim of the glass. Surface tension keeps the water from spilling over. The surface tension in the water is just too strong for the water film to stay intact as it stretches. Think about a piece of taffy - if you stretch it too far it breaks. Thus, to make a bubble we need to add to water some substance that decreases surface tension.
Soap is an ideal substance because it is made of long chains of carbon and hydrogen atoms.
This latter end likes to attach to grease. That is why soap cleans so well. At the surface, they tend to separate the water molecules, thus decreasing their attraction and reducing surface tension, typically to about a third that of water. Then, put your finger in soapy water and lightly touch the surface of the water. The water will immediately rush over the top due to the decrease in surface tension.