Sorry, can you give the source of the quote?
I do not believe there is any complete Tipitaka from as far back as the 15th century. In fact I have not seen any evidence of a complete Pali Tipitaka before the 5th Council edition of the late 19th Century. There may well be earlier complete texts, but in general, even the texts from the 18th and 19th centuries consist of specific manuscripts or sets of manuscripts.
Such surveys as have been done suggest that it was rare for a Theravadin monastery in pre-modern times to have a complete Tipitaka. Presumably the large central monasteries would have had complete collections, but I don’t know if any early ones have survived. If you see the ola leaf manuscripts in the Sri Lankan monasteries, they tend to be just gathered and piled together, not kept in an organized set.
What the quote is, I think, alluding to is the fact that from the 15th century, or better, from the 13th century, we start to have manuscripts that form part of a somewhat coherent manuscript tradition, as opposed to random scraps like the Nepalese fragment. Individually these manuscripts consist of parts of the Pali canon, typically a book of the Vinaya, a nikaya, or somesuch. Taken together they gradually accumulate the whole canon.
In the earliest manuscripts from the 13th/14th centuries, three main texts have been reported:
- Vinaya Cullavagga
- Vinaya Mahavagga
- Samyutta Nikaya
But I have only confirmed the Cullavagga.
From the 15th century onwards, manuscripts of the rest of the texts would gradually appear.