Being irrecoverably behind schedule does not constitute a failure unless you have an immutable deadline, like a launch mission to Mars whose launch window only happens once every two years. Otherwise, falling behind on schedule usually just means re-planning the project (e.g. increasing resources, reducing scope, or extending delivery date) and getting project sponsor approval for the re-plan.
A project is a failure when a decision authority says so at a review milestone. Every review milestone provides a decision authority with an opportunity to decide among several options, typically along the following lines:
1. Pass (or "Success") - proceed with the next phase of the project
2. Pass with actions - proceed with the next phase once several conditions are met, usually action items assigned to specific individuals or groups with specific due dates to keep the project on track
3. Revisit - go back and re-execute previous project phase(s) to resolve unmet objectives, outstanding issues, or unbearable risks, then come back and reschedule the review milestone again
4. Hold - stop work on the project pending some external condition (e.g. funding not currently available for the next phase, etc)
5. Terminate - do not proceed to the next phase but begin planning the liquidation of the project assets and re-coup of budgeted funds
It's this last option that means, "project failure."