Is there a point at which it's too late to get stitches?

Generally, About six hours after injury but must consider type of wound, location, age, health, how heavily soiled, etc.
Usually 24-36 hours. Most doctors are reluctant to close wounds after 24 to 36 hours. They believe that the risk of getting a wound infection begins to increase after that time period. Some areas can be closed later, like the face, because it is much less likely to get infected and closing with stitches has a better cosmetic result.