Short Answer
Often, yes. But compatibility and surrounding shingle condition should be considered.
Detailed Explanation
A single damaged shingle can often be replaced without disturbing the surrounding roofing. The process involves carefully lifting the shingles above the damaged one, removing the damaged shingle, cutting a new shingle to size, sliding it into place, and fastening it. However, several factors affect feasibility: the surrounding shingles must be flexible enough to lift without breaking (brittle shingles on older roofs may crack), replacement shingles should be available and reasonably matched in color, and the surrounding shingles should be in good condition (if the one next to it is about to fail too, replacing just one doesn't help much). On a relatively new roof with isolated damage, single-shingle replacement is a practical, cost-effective repair.
Why It Matters
Single-shingle replacement can be a cost-effective repair for isolated damage, but it's not always practical on older or deteriorated roofs.
When Repair Makes Sense
Damage is isolated to one or a few shingles, the roof is relatively new, and replacement shingles are available.
When Replacement Makes Sense
Damage is widespread, shingles are brittle and will break during repair, or the roof is at end of life.