Can I drill a 3 inch hole in a 2x6 stud to run wires?

Not without a protective plate, and not without checking the structural side of it separately. A dimensional 2x6 stud is actually 5.5 inches deep, and NEC 300.4(A)(1) requires at least 1.25 inches of edge distance from the nearest edge of the wood member to the near edge of the bored hole. On a 5.5 inch deep stud, a hole centered in the middle leaves 2.75 inches on either side before you eat into that required edge distance, so a hole larger than roughly 2.75 to 3 inches centered in the stud starts crowding or violating the clearance depending on exact placement.

When you can go bigger

If the hole cannot maintain 1.25 inches of edge distance on both sides, or if you are closer to one edge than the other, 300.4(A)(1) requires a listed steel plate at least 1/16 inch thick covering the area where the cable passes, sized to protect against nail and screw penetration. This is common where a hole is bored closer to the edge for framing reasons or where a larger diameter cable bundle needs more clearance than the centered hole allows.

The structural question is separate

NEC clearance rules protect the conductors from mechanical damage, they say nothing about whether a 3 inch hole compromises the load-bearing capacity of the stud itself. Building codes and engineered lumber manufacturer specs, not the NEC, govern maximum hole size relative to stud width for structural framing members, and a bearing wall stud has tighter allowances than a non-bearing partition. Check both rule sets before drilling a hole that large.

Field takeaway

Keep 1.25 inches of wood on each side of the bored hole per 300.4(A)(1), or install a nail plate if you cannot. Separately confirm the hole size does not exceed what the framing code and stud type allow structurally, especially on load-bearing walls.