We study the stability of the cohabitation of French and Dutch in the Brussels-capital region (Belgium). To this aim, we use available time series of fractions of speakers of monolinguals of both tongues as well as the fractions of bilinguals. The time series span a period from the mid-XIX century until 1947, year of the last accepted linguistic census. During this period, French penetrated the Dutch-vernacular region of Brussels and started a language shift that lasts to this day. The available time series are analyzed with a mathematical model of sociolinguistic dynamics that accounts for cohabitation of languages along bilingualism. Our equations are compatible with long-term coexistence of both languages, or with one tongue taking over and extinguishing the other. A series of model parameters (which we constrain by fitting our equations to the data) determine which long-term trajectory (cohabitation or extinction) would be followed. This allows us to estimate whether the empirical data are compatible ... : 10 pages, 5 figures ...