What will happen if water is compressed more and more? Can its bonds break in this way?
You can answer this question by looking at the phase diagram for water. A phase diagram is a chart that shows the relationship between temperature, pressure, and the solid, liquid and vapor phases of a substance.
The answer to your question depends in part on how high of pressures we're talking about, and in part on the starting conditions of the water. If the water is already liquid, then compressing the water will never create ice because water is one of the few substances that has a vapor/solid phase line with negative slope. This also means that if you start with ice near the phase line, increasing pressure will actually produce liquid water.
However, if you get to extremely high pressures, like around 10^9 Pa, the water enters a plastic-crystal phase, which is a weak solid state.
The answer to your question is yes, if you continue to compress the molecules, the temperature will eventually be such that the bonds break. As you increase pressure, temperature increases, and there is a practical limit to the ability to artificially cool the sample.