The individuals to which @Sicarius referred in the UK came from the population at Al-Bustan Zoo in the UAE, which has existed for around 15-20 years by now; their deaths were due to unconnected illnesses, rather than the species not living long in captivity.
Even as long ago as the 1970s and 1980s the captive population then present in Europe and North America often lived similar lifespans to other small cat species - for instance, one male born at Los Angeles Zoo on 23 August 1976 lived over 13 years in captivity, dying on 31 January 1990 at Cincinnati Zoo. This is a similar lifespan to the maximum captive longevity for Flat-headed Cat (a wild-caught individual born in c.1967 which arrived at Lincoln Park Zoo the following year, and remained there until it died on 9 November 1981).
The issue, I suspect, is more that the two species never established themselves as long-term breeding populations in zoological collections, and nowadays (with the aforementioned exception of the marbled cats at Al-Bustan) tend to occur as isolated individuals in Asian collections.
As for Bay Cat, precious few have *ever* been held in captivity - even in Asia - for any educated judgements to be made about their captive longevity; the sole recent example, at Taman Safari Bogor, was only on-display for a relatively brief period before dying of old age - but it transpired that the cat in question had actually been present at the collection off-display for around a decade, having been misidentified as an Asian Golden Cat.
