The reason I ask this is because I am researching on the background of a common phrase used today. My teacher says that the phrase can be tied to events from Medieval England, but when I researched it, I keep getting that it originated in Roman times.
In just about all situations, they're exclusive. What's the context of this? It's possible your teacher or your sources are wrong.
Historical eras are not categories that are set in stone, and for the most part, are fairly arbitrary. Well, that is to say, broad eras such as "classical antiquity" and "the Middle Ages" are arbitrary (more specific time periods - such as Tudor England or Edo Japan, can obviously be defined). As my high school history teacher said, people didn't just wake up one day and realize that they were now living in the Middle Ages.
That being aid, these eras exist in historiography because they can be convenient. People shouldn't stick to them religiously, but they can be useful for splitting up and studying the past.
When most people talk of "Roman times", they are talking about an era known by such terms as classical antiquity. This is the ancient world in which the Roman Empire rose and flourished within. Historians date the end of antiquity differently (and as aforementioned, these ending dates really are arbitrary), but antiquity is not seen as being the same as the Middle Ages (or Medieval period, the two terms refer the same thing). If antiquity and the Middle Ages were the same, then the two different labels simply wouldn't be useful.
In terms of "Roman times" as in "the Roman Empire", well, Roman rule ended in Britain by a little after 410 AD. Romano-British culture continued to exist, of course, but Medieval England was not Roman in the sense that it was not a part of the Roman Empire. The Roman Empire did survive in the East - today we often use the term "Byzantine Empire" to describe the Eastern portion of the Roman Empire that survived until 1453, but the the residents of the Empire really did see it as the Roman Empire.
No.
Simply because the level of which two completely time periods with completely different everything could be considered "the same" would be a level of reduction that would be completely absurd, akin to asking "can the Civil War and WWII be considered the same?"
If you reformulated your question to say "could be considered similar," then the question becomes more viable (although still tremendously reductionist, and still mostly wrong in both structure and detail).
However, the late antiquity period is commonly considered a "pre-cursor" to the (early) medieval period, sharing some overlaps (the rise of mounted soldiers, fortified strongholds in lieu of open cities, an aristocracy whose wealth was based on land rather than exchange or capital), but even these comparisons tend to break apart upon close examination.
That a phrase or a concept from Medieval England originated from Roman times, does not mean it is the "same." It just means it originated from Roman times.