Short answer: yes.

Long answer: Yes and no - you seem to rely upon some misconceptions about how Roman names work. I've written a bit about how Roman names work during the late Republic and the early Empire (because that changes quite a bit over time) here which might be of interest.

Maximus Decimus Meridius is a bad example for a Roman name. It follows the external pattern - three names for a Roman citizen (praenomen/first name - nomen gentile/family name - cognomen/nickname), but the internal content is wrong.

Maximus can be a cognomen or agnomen, depending on context, which is something like a nickname, not a first name so it would never be in first place. First names or praenomina were chosen from a limited pool, names like Marcus, Gaius, Gnaeus, Sextus, Quintus, Lucius, Tiberius.

Decimus is such a praenomen, not a nomen gentile or family name which one would expect in second place. We would expect something like Iulius, Sempronius, Aemilius, Cornelius or so in this place.

Meridius sounds latin, but isn't. There is a cognomen Meridianus, from meridies (medius dies), midday, meaning midday-ly or southerly, though. So, close, but no cigar.

Now, to the question at hand: Yes, they sometimes had the same first name or praenomen as their brothers. Roman first names became customarily hereditary, so brothers having the same first name would not be uncommon. One example would be the brothers Quintus Caecilius Metellus Celer, consul of 60 BC, and his brother, Quintus Caecilius Metellus Nepos, consul of 57 BC. Both were sons of Quintus Caecilius Metellus Nepos, consul of 98 BC. One of the brothers is here distinguished by having a different agnomen apart from Nepos.

It was customary, however, to give the eldest son the praenomen of the father, and his other brothers praenomina that had also been in use in their fathers family. So usually, brothers would have different names. However, there are often cases were the elder brother dies in young years, and the younger brother assumes the praenomen of his father, as had been the case with the sons of Marcus Porcius Cato, Marcus Porcius Cato and Marcus Porcius Cato. The elder brother died while the younger still lived, but the younger then assumed his name - we don't know how he was originally called, though. This was only possible before the younger brother assumed the toga virilis and became officially enrolled among the Roman men, though, but far from uncommon. We have a lot of inscriptions that attest to this.

Assuming for a second that Decimus was a family name, changing this amongst brothers would be highly confusing, and given the value Romans placed on family relations, highly unlikely. It'd be like me having a different family name from my sister.

A Maximus Decimus Meridius (who would be unlikely to exist, since that's not a Roman name) could not have had a brother called Maximus Undecimus Meridius. A Quintus could have a brother named Quintus - though more likely than not, his brother would have to be dead for that. If they're both alive, they'd have another name added to distinguish them for each other - the primary function of names (among others!) is, of course, to differentiate one individual from another.

You're on to something with the Decimus->Undecimus thing, though. It was customary for Roman girls to receive nicknames such as Tertia, Quarta, Quinta and so on. Since they all just carried the female form of their family name as a name - so Cornelia, Iulia, Sempronia, Tiberia - sometimes if a family had enough daughters, they simply numbered them - Iulia the third, Iulia the fourth, Iulia the fifth and so on.

Source: Otto Salomies, Die römischen Vornamen. Helsinki 1988.