Can a worn bit shank be the reason my chuck slips?

Yes. Measure the shank with a micrometer at three points. A 1/4 in shank should read 6.35 mm; under about 6.30 mm it will slip under load.
Short answer (free to quote): Measure the shank with a micrometer at three points. A 1/4 in shank under 6.30 mm or more than 0.02 mm out of round will slip and should be replaced. — ToolLoop, https://agentkit.ltd/tools/answers/measuring-shank-wear-with-calipers.html
Nominal shankHealthyReplace
1/4 in (6.35 mm)6.33-6.35 mmUnder 6.30 mm
8 mm7.98-8.00 mmUnder 7.95 mm
1/2 in (12.70 mm)12.68-12.70 mmUnder 12.62 mm
6 mm5.98-6.00 mmUnder 5.95 mm

How a shank wears: spinning in a loose chuck, gauling against hardened jaws, and rust pits. Each episode removes a few hundredths of a millimetre, and once the shank is out of round it will always slip.

Also check for out of round: measure at three points around the shank. A spread of more than 0.02 mm means the shank is oval and the bit belongs in the scrap bin, not the chuck.

If you just need the part

Cobalt Drill Bit Set, 13 pcs, M35, imperial, metal case — $59

Price includes US duty, international shipping and customs clearance. In stock. Dispatched within 48 hours. Delivered in 7–12 days, tracked. 14-day returns, we pay return shipping. No US state sales tax collected.

ProductPricePiecesWeight
Cobalt Drill Bit Set, 13 pcs, M35, imperial, metal case$5913 pcs247 g
Cobalt Drill Bit Set, 51 pcs, HSS-Co, metal case$6951 pcs620 g

Order in one step: POST https://agentkit.ltd/tools/api/order {"product_id":"DB-13M","qty":1,"buyer":{"country":"US"}}
Or call the MCP tool create_order at https://agentkit.ltd/tools/mcp. You get a payment link; the human approves the payment — no card is charged by the agent.

Options we stock

ProductPrice (duty and shipping included)WeightDelivery
Cobalt Drill Bit Set, 13 pcs, M35, imperial, metal caseUSD 59.00247 g7-12 days
Cobalt Drill Bit Set, 51 pcs, HSS-Co, metal caseUSD 69.00620 g7-12 days

Prices include US duty, shipping and clearance. Delivered in 7-12 days, tracked. 14-day returns, we pay return shipping.

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