In Time vs. On Time

The expression in time is often followed by an infinitive:

We were late, but we still arrived in time to see the beginning of the show.

When it is not followed by an infinitive, an infinitive is still implied by the context:

No, we didn’t get there in time. (In time to do what? In context, the implication is that we didn’t get there in time to see the beginning of the show.)

The expression on time refers to a time scheduled in advance and implies punctuality:

We arrived at the theater on time. (We arrived by the time the movie was scheduled to begin.)