I came across a post yesterday, which, sadly, I cannot find now, that claims that Basque and Celtiberian were spoken in Spain even after the Western Roman Empire collapsed. Latin only became commonly spoken after the Arab invasion, with the remaining Visigothic nobles imposing the Latin language on the commoners. I have a strong suspicion that this post was wrong, but I want someone with knowledge to shed some light on the subject.
The paleohispanic languages, except for Basque, were replaced by Latin from 200 BC - 100 AD. The Latin spoken in Iberia was already changing towards Castillian, Portugese, Gallician, etc. by the time of the Arab invasion.