To have many responsibilities or tasks to handle; to be very busy.
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To have many responsibilities or tasks to deal with at once.
When you have a lot on your plate, you are very busy and stretched thin with commitments.
It is used to explain why someone is overwhelmed or cannot take on anything extra.
"Stretched Too Thin"
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Spread across too many tasks so that quality or focus suffers
"In Over Your Head"
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Involved in something too difficult or complex to handle
"Have a Lot on One's Plate"
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Having too many responsibilities or tasks to manage at once