I've developed a high tolerance for a drug. Does this mean I'm less likely to overdose?

No. It may mean you are more likely to overdose since it will require more drug to achieve the same desired effect. The best way to avoid overdose is not to take the drug in the first place. Believing one is safe from overdosing is the fastest way to end up overdosing.
Vague question. What kind of a drug are you talking about? Different drugs have different mechanisms of elimination and action and not all of the drugs would cause an overdose. High tolerance usually does mean, however, that an overdose is less likely with usual concentrations, but never impossible. It just depends on the dosage.