The honest starting point
If allergies are the deciding factor, a Japanese Bobtail should not be sold to you as a safe bet. The proteins that trigger most cat allergies come mainly from skin and saliva, not from hair itself, and every cat produces them, including this breed. A Japanese Bobtail grooms, sheds, and spreads those proteins around a home just as other cats do, so a person with a confirmed cat allergy may react to one exactly as they would to any cat. Where the breed offers a small edge is in coat volume. The Japanese Bobtail carries a single coat with little undercoat, in short or long varieties, and sheds moderately rather than heavily. Less loose hair drifting onto sofas and bedding can mean less allergen being carried around the house, which is why some mildly sensitive people find life with a light-shedding cat manageable. Manageable for one person, however, says nothing certain about the next.
Why individual cats differ so much
Allergen output varies from cat to cat, sometimes more between two individuals of one breed than between breeds. Age, sex, and the particular cat's skin and grooming habits all shift how much reactive protein ends up in the air and on surfaces. This is why blanket breed lists labelled hypoallergenic mislead people, and why your own experience with a specific Japanese Bobtail is worth far more than any general claim about the breed. It also means the kitten you visit is not the whole story. Young kittens often produce less reaction than they will as adults, so a comfortable afternoon with a litter of Bobtail kittens does not guarantee a comfortable year with the grown cat they become.
Test with time, not a quick cuddle
Arrange real exposure before committing. Visit the breeder's or shelter's cats more than once, stay for an hour or more, handle the cat, let it sit on your lap, and then notice how you feel that evening and the next morning, since reactions can build slowly. Rubbing your face after handling the cat, however unglamorous, tells you more than petting with fingertips does. If the household member with allergies is not the person choosing the cat, that person must be the one who tests. A partner reporting that the cat seemed fine is no evidence at all about the sufferer's actual response.
Living with a Bobtail when allergies are mild
For people whose reactions are mild and confirmed manageable by their own doctor, daily habits decide a lot. Keeping the cat out of the allergic person's bedroom and off their pillow protects sleep, which is when symptoms bite hardest. Weekly wiping of the coat with a damp cloth, regular vacuuming of soft furnishings, washing hands after handling, and good ventilation all reduce the background level. Grooming helps this breed anyway. A Japanese Bobtail's silky coat benefits from a comb-through once or twice a week, and doing it in one spot, then cleaning up, concentrates the loose hair instead of spreading it through every room.
Making the decision with clear eyes
Adopting a cat you may not be able to keep is unfair to everyone, including the cat, so treat the trial period as the decision, not a formality. A good breeder or rescue will welcome the questions and may agree a trial arrangement; reluctance to discuss allergies at all is itself useful information. Before you commit, put a follow-up question to KinsLe's Kimmy about what your trial visits showed; she answers using the KinsLe Answers library.