  The use of the adiabatic approximation in practical applications, as in
adiabatic quantum computation, demands an assessment of the errors made in
finite-time evolutions. Aiming at such scenarios, we derive bounds relating
error and evolution time in the adiabatic approximation that go beyond typical
scaling relations. Using the Adiabatic Perturbation Theory, we obtain
leading-order expressions valid for long evolution time $T$, while explicitly
determining the shortest time $T$ and the largest error $\varepsilon$ for which
they are valid. In this validity regime, we can make clear and precise
statements about the evolution time needed to reach a given error and
vice-versa. As an example of practical importance, we apply these results to
the adiabatic search, and obtain for the first time an error-run-time trade-off
relation that fully reproduces the discrete-Grover-search scaling. We also
pioneer the obtention of tight numerical values for $\varepsilon$ and $T$ under
the error-reducing strategy ``boundary cancelation''.
