Both the bar magnet and the field around a wire COULD be cancelled i they were infinitely thin (an impossibility). Neither can b perfectly cancelled in real life because they are of finite size. Perfect cancellation could only be achieved by exact superposition o all elements of the wire or magnet with an element of the opposit polarity/current direction at the same point in space. Since you ca never achieve such exact superposition due to the physical size of th wire/magnet, you can never achieve perfect cancellation - onl approximate cancellation, for example, by placing two small wire very close together. Twisting two wires helps by averaging th imperfectly cancelled fields from the two wires, but this is als just an improved, imperfect cancellation
After all, the absence of a magnetic field around a non-magnetize
piece of iron is due to the micro-scale mutual cancellation of al the randomly oriented tiny magnetic domains of the iron. They cance "perfectly" on a macro scale because the domains are so small and w are normally sensing the average field of all those randomly oriente domains. But if you could probe with a micro probe, you would be abl to sense the magnetic field around each minute magnetic domain and yo would see that, at that scale, the cancellation is not perfect due t the non-infinitesimal size of the domains
Does this help, or just add to the confusion
awrigh