I've looked at this question from a couple of different angles, the first of which is specifically about the name "Thomas" and opens with--I promise I did not just edit this in--the three Catherines, two Annas situation.

I'll paste the later answer here because's it more of an actual answer to your question, although my examples are German:

Yes! I love this question so much. When you look through the letter collection of Hildegard of Bingen in the late 12th century, her epistolarly correspondents and the people she mentions have names like Tenxwind (f), Richardis (f), Helengarus (m), Kuno (m). The only real trouble you'll have with overlapping names is keeping the various royal Henries in line. When you look at 16th century cities, the most popular boys' name is Hieronymous (Jerome) after the late antique philologist-theologian and everyone else wants to name their kid Desiderius after Erasmus.

In between, the boys are all named the local version of John, Thomas, and Peter, and the girls are are Elisabeth, Mary, Margaret, Katherine, Anna, or Barbara.

There were two major forces at play in the homogenization of first names (because this is also the era when personal names were becoming "first" names"), both of which stem from the deeper Christianization of the laity of the course of the later Middle Ages.

The first trend and underlying reason is the shift towards saints' names, and in particular, a certain subset of saints. Apostles' names stay very popular among men. We should think of these as coming out of the stories told about Christ's life rather than "bible stories," specifically--a wider use of biblical names is more associated with the Reformed/Puritans later on.

With women, the so-called "virgin martyrs" of late antique legend achieve rockstar namesake status: Katherine (of Alexandria, defeated 50 pagan philosophers in oral debate), Margaret (of Antioch, the dragonslayer saint), Barbara (the Rapunzel saint). The "life of Christ" tradition also draws out a few: Anna, Elizabeth, Magdalena, and even Maria (although the last is not as common as you might think--the Catholic tradition of naming every daughter Mary _____ is rather modern). The popularity of Elizabeth, too, gets a boost from St. Elisabeth of Hungary in the 13th century, who became one of the two most popular saints among medieval women and especially religious women in the late Middle Ages (Katherine of Alexandria also represents well).

This is not a perfect rule, and variations are often regional. In Germany, the women's names of Adelheid and Gertrude remain tenaciously popular, for example, and I don't think you can put all the weight of those on 7C abbess St. Gertrude of Nivelles. Not sure where the popularity of "Conrad" came from, but presumably it wasn't anywhere good. The three Konrads that spring immediately to my mind (of Marburg, Kügelein, Peuttinger) were each despicable in some way (ALL the ways, no seriously all of them, everyone hated Conrad of Marburg; creeper on young virgin women; Toddlers & Tiaras dad). One assumes there were at least some Conrads who turned out normal.

The turn towards saints' names related to the rising presence of hagiographic legends (including the vita Christi genre) in later medieval culture, and possibly some superstition about baptizing a child with a saint's name being god for their salvation.

But there's a second step to consider in the winnowing down of most western Europeans' names--obviously there would always be some wonderful exceptions--to such a small group. It also relates to baptism, and it's the central role that godparents played in later medieval culture.

Godparents as well as parents stood up for the baby at baptism, with all the same religious and upbringing requirements. In fact, the godparent relationship was considered so strong by medieval people that it counted for rules on incest and consanguinity.

(Tangent: also the "incest taboo" is an anthropological commonplace, one of the interesting points that researchers make is that every culture defines what is "incest" somewhat differently, beyond the parent/child taboo. In medieval Europe, baptismal parents and the "familial" relationships established from that are a fascinating demonstration of this).

The importance of godparents and the connection with baptism, it seems, helped lead to a cultural fad of naming the baby after the godparents. And so one generation tended to pass its names down to the next.

Obviously, none of these are exact rules. The slate of common names is a little broader than I made it out to be (of course). And then there's always going to be the mom who insisted on naming her daughter Isolde after the Arthurian romance. Nevertheless, it's an interesting exercise to go through the "Personenregister" (people-index) of medieval letter collections or studies of convents and write down the occurrence of contemporary names. It's pretty revealing.

A user in the other thread suggested that grandparents as well as godparents may have been a factor, but this is not generally something I see in my own sources. The role/presence of grandparents is harder than one might think to trace in the late Middle Ages. Although it's also possible that reflects a lack of attention in some cases, not just lost or never-written records. Albrecht Durer's biographers have suggested his parents chose godparents for their children based on what they expected the name to be, so it could be an indirect factor in some cases.