Mostly not. The austenitic nickel alloys and the nickel-copper grades are non-magnetic in the annealed condition, which is why Monel appears in instrument and sensor work. Grades with a substantial iron content, such as 718, are noticeably magnetic, and heavily cold-worked parts can show weak magnetism even in normally non-magnetic grades.
It follows the structure, not the name
Magnetism depends on whether the crystal structure is ferritic or austenitic, and on how much iron and cobalt are in the alloy. Austenite is non-magnetic; ferrite is not. Nickel alloys with a fully austenitic structure — 600, 625, C276, Monel 400 — are essentially non-magnetic.
718 is the obvious exception in the Inconel range. It is a nickel-iron-chromium alloy with roughly 20 % iron, and it responds to a magnet in a way that 625 does not.
Two effects that surprise people
Cold work. Drawing or bending a normally non-magnetic austenitic grade can induce a little ferrite, or align the structure enough to raise permeability. A deep-drawn 316 part may show a weak attraction that the same material in the annealed condition does not.
Heat treatment. Ageing a precipitation-hardened grade can precipitate phases that change the magnetic response, so the condition on the certificate matters as well as the grade.
When magnetism is the requirement
For instruments, sensors, MRI-adjacent hardware and anything near a magnetic field, specify the permeability limit you need, not just the grade. The soft-magnetic precision alloys — the 1J50, 1J79 and 1J85 type grades — are bought precisely because their permeability is controlled and specified.
For everything else, treat magnetism as a property to confirm rather than assume, and ask for it to be stated on the test report if the end use cares.