The phrase usually carries a critical tone, suggesting someone's behavior is inappropriate or arrogant. He's been too big for his boots since he won the award.
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Informal Usage
It's often used in informal settings, such as personal conversations, not in formal or professional contexts. She's really starting to get too big for her boots with her friends.
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Personal Success
Typically used when someone lets a small personal success affect their attitude towards others. They've been acting too big for their boots after the minor promotion.