Is it likely to have a stable relationship and have borderline personality disorder?

It's possible: One of the hallmarks of bpd is unstable relationships. But like most matters in psychiatry, it isn't 100%. It's entirely possible for someone with bpd to have a stable relationship, especially with someone who is consistent and trustworthy. Also, bpd varies in severity. There are mild cases that are hardly recognizable as bpd, and where stable relationships are not only possible but likely.

Borderline: Absolutely. Staying in therapy & having strong social support system are vital.

Variability: I agree with my colleagues about the range of illness severity, and the helpfulness of dbt. If the person is committed to both self-acceptance and need for change, the storms in relatedness may gradually quiet. Without intensive therapy along w/dbt, the suffering inside the bpd person can be so intense that perceptions are very distorted, and the partner's emotional reserves are used up too.

Borderline: Not likely at all. One of the hallmarks of BPD are disturbed interpersonal relationships, gross emotional reactivity, as well as intense fear of abandonment, self-mutilation, manipulativeness, thinking of people/things in all good or all bad terms, and a profoundly disturbed sense of self.