Although the Balkan-Danubian linguistic situation is sticky, the Italian peninsula of pre-history and early history is even worse.
Here's the obvious ones: Latin is an "Italic" language. The Italic family comes in two varieties, the Latino-Faliscan and Osco-Umbrian. Latin, obviously, belongs to the Latino-Faliscan family and eventually pretty much swamped all the other languages on the peninsula and beyond. But before that happened, Latin was isolated to Rome and the surrounding countryside and tons of other languages competed for supremacy.
Etruscan and the related Rhaetic held forth up north. These were non-Indo-European languages belonging to the Tyrsenian or Aegean language family, which is sadly lacking in sufficient evidence to really be more than a hypothesis. North Picene is another non-Indo-European language (in contrast to South Picene, which belongs to the Osco-Umbrian group). Camunic, Sicel, Elymian and Siconian are all of dubious classification; maybe Indo-European (of an undetermined branch) or possibly not even that.
Messapic is actually thought to be an Illyrian language (or close relative) that crossed the Adriatic. Lepontic is believed to be a Celtic language and Ligurian is believed---possibly---to be one as well. Vestinian is believed to be Indo-European, and Venetic is definately Indo-European, although it's affinities are unclear (correspondances with other Italic languages, Illyrian languages and even Germanic languages have all been remarked upon.) Of course, Greek is also attested, since before the rise of the Romans, Greek settlers and colonists on the Italian peninsula are also known.
All this discussion about the diversity and complexity of the linguistic picture in the Balkans and the Italian peninsula on the very early days of writing in Europe lead me to believe that our picture of linguistics in other areas is probably over-simplistic. Why would these areas be so linguistically diverse while to the north, northwest and northeast only a few languages developed (Celtic, Germanic, Baltic and Slavic)?
The answer is; they were probably just as diverse too, at one point, but one language achieved some kind of dominance and gradually drove the others to extinction. That's exactly what happened in Italy; it took a few centuries, but gradually Latin and it's descendents and dialects wiped out all of its competitors. Because Italy was literate earlier than, say, Germania or Gaul, who's to say that those territories weren't just as linguistically diverse before proto-Celtic or proto-Germanic swamped them? In fact, some linguists have proposed that there is some circumstantial evidence of a culturo-linguistic entity dubbed "Nordwestblock" that existed between the Celts and the Germans, mostly occupying what are today the Low Countries.
Historically, we also saw the same thing happen in the Balkans; Greece, Phrygian and Armenian left the area to go on to survive (and Anatolian, if that's where that language came from) and although Illyrian, Thracian and Dacian are not well known languages to linguists today, we at least know they existed. Today, the advent of various Slavic languages has almost completely swamped that prior diversity, with the exception of Albanian and the unrelated interloper Hungarian, which also arrived in historical times.
If you look at the Caucasus, we see a similar degree of diversity in a small area, with multiple language families and a few language isolates. Ancient Anatolia also had similar linguistic diversity with several Anatolian Indo-Hittite languages competing with Semitic, Hattic and Hurrian languages, as well as the linguistically obscure Kassites. The Tarim basin at the start of the common era was similarly diverse; two varieties of Iranian language were common (Khotanese Saka and Sogdian) probably three Tocharian languages, Chinese, several Indic languages (Prakrits, and Sanskrit as a religious liturgical language) as well as various other Turkic and/or Siberian languages making inroads from time to time (all of the above were swamped by Uyghur later, and modern Chinese is making serious inroads even into that.) For much of that time, people in the area were multilingual and used different languages every day depending on what social domain they were interacting.
The point is, without the impetus of some kind of linguistic spread (conquest by outsiders, trade pre-eminence, frequent travel contacts, etc.) linguistic diversity is normal. As people settle into their little areas and tend not to leave, they rather quickly "speciate" into a bewildering variety of dialects, which go on to splinter into different languages. Even cultural dominance, as by an empire or something, doesn't last forever. Latin was already breaking up into regional dialects before the Western Roman Empire finished falling; the Eastern Roman Empire adopted Greek, and the vulgar Latin broke down into Spanish, French, Italian, Portuguese, Galician, Aragonese, Catalan, Provençal, Romanian, Moldavan, Occitan, and more in fairly short order, and even today with the advent of more powerful nations and nationwide media, much of that diversity still exists in Europe. Similar linguistic diversity can be seen in the remnants of the old Persian Empire as well. Or China. Or India.
As a person who enjoys developing fantasy settings for writing and for gaming, this is an interesting phenomena. Few such setting creators go for linguistic diversity. It's difficult, for one thing. It creates problems. It's not fun for gamers or readers to have characters unable to communicate. A common solution (if you'll pardon the pun) is the adoption of a "Common" tongue. Tolkien can probably be credited with this idea; his Westron (more commonly known as "Common) was the language du jour for almost the entire region in which Lord of the Rings takes place. Even in places where relict languages still existed, enough bilingual people existed to facilitate communication, i.e. with the dwarves or the elves. D&D has a similar conceit; Common also being the name of their lingua franca, which is the native tongue of all humans in most settings.
Despite its ease, it's also unrealistic and unsatisfying to someone like me to whom linguistics is a bit of a hobby. I'm not sure I understand or have decided exactly where the dividing line is between "enough" and "too much" linguistics in my fantasy, though.
Any comments from the peanut gallery?