The simple answer is, yes. The Xbox One has many features that require more bandwidth such as video and music streaming (as most content is served in higher resolution). However, these services are not required to game.
Services such as chat might require higher bandwidth but only marginally as Codec's, what is used to encode and decode the voice before sending it, allows for higher quality sound to be compressed into less data.
By your description I believe that you do not fully grasp the reason why you are lagging out. Indeed, lag is linked to bandwidth but it is usually not the deciding factor. What is more important is latency- Please check a site such as Speedtest.net to see what your latency is.
Latency is the round trip time of data you send out. Basically, imagine it as a postal service. Latency is the time required for a letter you send to be responded to, i.e, when you hold the reply in your hands and can read the response. This is a very important concept as online servers rely on the responsiveness on all participants. So when the host server detects that you take to long to respond to the packets it sends your way, it will drop your connection to avoid impact on the quality of the game for other players. That is why you lag out.
Therefore, a low latency is what you want. However, latency is only loosely tied to bandwidth. It has to do with the quality of your equipment, if you are on wireless or wired, your ISP, your geographical location, and the location of the remote server. There are more factors that are basically related to the above. So when you play on a server hosted on a different continent, your latency will be higher. That is normal and expected.
Now, sudden drops in latency are usually what cause sudden lag or drop-outs. They are caused by bursts of data that clog up the "pipe". Again, imagine it as a postal service. While on a normal day they can receive and forward your mail in a few hours, during Christmas season they are swamped and will take longer. This is to be expected. Similarly, if you are saturating your connection or equipment with downloads, streams, or similar, your equipment, especially older equipment, might struggle to forward all packets in time and connections might drop.
The tricky part is that there is not always something you can do. If you live in an apartment building that shares a physical line, the other tenants might have some activity that is transparent for you but still impacts your connection. The same might hold for your street as all connections on a street usually share the same physical line which uses the same physical equipment down the line.
One way to go might be to contact your ISP and ask what you can do. They should be able to help you. Personally, I'd advise you look into getting on a connection based on optical fibre. They feature less latency and can carry more bandwidth while alleviating some of the problems I mentioned above. If you want you can take a speed test and post it as a comment, I'll try to give you more streamlined advice.