It would depend on whom you were speaking to. You would probably be mostly understood in a court of law or among polite society as far back as the 1500s. However stopping a random person in the street for directions, unless the person you happened upon was university-educated, you would have serious difficulty making yourself understood to anyone prior to the advent of radio in 1924.

Modern Italian is a constructed language. It is what a clique of 19th-century politicians and intellectuals believed was the most understandable form of the language spoken in Florence, itself chosen for its historic links to a large body of "Classic" medieval poetry which had long been established as the most universally comprehensible standard (a phenomenon which sometimes perplexed the Tuscans themselves). But while modern Italian is derived from the Florentine variety of Tuscan, it is ultimately very different from it. The clearest example might be the famous "Tuscan Gorgia," whereby tuscans convert voiceless stops like /k/ /t/ and /p/ into a sort of /h/ sound, a phenomenon entirely absent from Italian (or indeed any other dialect on the peninsula).

While we speak of "Italian Dialects," it is more correct to say "Dialects of Italy," since each region (and sub-region) of Italy experienced a parallel and unique divergence from Latin. Further, insistence on the use of Latin as a "Prestige" language led to centuries of poorly documented local dialects drifting further and further from each other and from Latin, with no connotations of a unifying "Formal Register" to bind them all together. Indeed, the spectrum from local vernacular and Latin could be variable, with the most "Formal" or "Prestigious" register would be Latin which allowed the fewest infiltrations from whatever local dialect was native to the speaker (much much of the same way "Formal Italian" is perceived nowadays).

The emergence of universally acclaimed Tuscan poetry and literature in the early 14th century would provide well-educated Italians with an alternative means of communication to Latin. Indeed, the Tuscan vernacular could quite possibly have been perceived as even more prestigious than Latin, as its non-Tuscan adopters could tacitly signal to each other they had read and understood the most popular literary and philosophical works of the day. The arrival of the printing press further sealed Tuscan's status as "Standard," as publishers quickly adopted it given it had already become the most universally-understood register among the literary class. As Tuscan replaced Latin it was also adopted as the language of diplomacy and courtly life, as aristocratic individuals from different parts of the peninsula came to expect knowledge of tuscan from their counterparts.

The widespread adoption of the "Tuscan Standard" did not pass without comment. Tuscan poets had established their primacy by being the first to write in their local vernacular, but by the time the printing press emerged it was Venice, and not Florence, which housed Italy's most florid literary scene. This meant that, somewhat bizarrely, Venetians were publishing many more Tuscan dictionaries and grammar primers than Florentines were (while some Venice-based grammarians were themselves originally Tuscan, many others were not). Grammarians in Venice (as well in other printing centers) also began modifying "Literary Tuscan" as its writers and speakers across the peninsula simplified the language (even though at this phase, its learners were still largely upper-class Italians). Some of these changes were adopted to reflect how more contemporary Florentines spoke, while others represented the slow adoption of northern-isms into this "Prestige" register (although Tuscan and Florentine literary circles made a point of largely ignoring the latter changes).

The ultimate use of "Italian" as a literally language adopted by a very small proportion of people meant it did not have many opportunities to change once it was formalized in early printed works. While some expressions, spellings, and grammar constructs might seem strange to modern readers, texts written as early as the late 15th century (immediately prior to the printing press "Freezing" the language) are largely comprehensible to modern readers (as you yourself noted).

But spoken Italian is a wholly different story altogether. I've passed the word limit, so I'm going to continue after the jump, in a reply to this post.