Yes, it happens. Most people who are going fail, do so within the first couple of years, but there are always a few people who fail latter years / Finals as well. I was almost in this situation, as I failed 4th year, but successfully appealed and continued. I had intercalated after 3rd year, and so already had a BSc. At my university, those that fail after 4th or 5th year, were awarded a BSc in "Medical Studies", as a recognition of the credits they had gained in previously years.
On one hand, this means that people leave with a degree, and therefore can go on to do any 'regular' graduate job that just requires a degree in something. On the other hand, it's not very useful for specific jobs - ie if someone wanted to pursue a career in healthcare science such as a lab scientist or a nurse, it's not like a biomedical science degree where there has been a lot of lab experience or a degree that allows any kind of professional registration. It does however mean that you don't have to start all over again just to get a degree in something. I don't think they gave the BSc to people who failed 3rd year, as technically, if you failed 3rd year then you didn't get all the credits from that year and you need three years' worth of credits to achieve a degree. When I intercalated (in a humanities subject), there was a dentistry student who had failed her third year, who had been allowed to transfer onto the intercalated programme so that she would be able to graduate with a degree.
I have no idea if there is a similar system at other universities.