If you make the bicycle fit, then the frame was not a size too small or a size too large. Too large and too small are sizes where the stem, handlebars, and seatpost that you want to use do not result in a good fit. However, there are a lot of frame size finding methods out there with a purpose to help buyers find the "right" size frame to buy. If a person determines their "correct" frame size using one of these methods, what will result from buying and trying to fit a frame a size smaller or larger than the method specifies?
Buying a smaller size frame on a modern road bike (last 25 years perhaps) will probably result in a lower bar position. These bikes tend to have a low stack-to-reach ratio. Even the small sizes are long-reach compared to vintage bikes. So the most consequential result is that you'll have a low bar position, which in some cases may be desirable. It can be counter-acted with a taller steerer or stem which can become problematic at extremes but should be fine if the change is only the 1cm or so between sizes. If the frame is small enough, the seat-tube angle can also be steeper to avoid toe-overlap. This often becomes a factor with sizes below 54cm, like 51cm or 48cm and so on. It may not be a big deal, but it is a difference.
Buying a larger frame, traditionally, with the level top-tube, the problem was excessive stand-over height. The rider might have to lean the bike over to dismount the saddle or risk the top tube impacting their groin. At some point, it also becomes impossible to lower the seat sufficiently as the seat cannot be pushed below the joint with the top tube. With modern road geometry, an excessively large frame may have too much reach. If we're talking one or two frame sizes, this can be counter-acted with a shorter stem or by pushing the seat forward on the post. These actions have some consequence in the steering and effective seat tube angle.
My road bike is a size smaller than the sizing methods suggest. The result is the bars are well-below the seat and a very aggressive riding position. I read that some pros ride a size or two small intentionally. I did it only because of a good deal on a used Bianchi. I did for a while bring the bars up to seat-height with a riser stem. That worked very effectively and I was able to ride 12 hours on it without discomfort. I've gone back to the original stem to sell the bike. It's probably better this way for shorter, aggressive rides, but I ride my touring bike more.
My (vintage) touring bike frame is two sizes larger than a road-bike sizing method would suggest. Again, I got it primarily because it was available and I liked it better than other available bikes. The stand-over height isn't excessive. I can straddle it with clearance. There is not a lot of seat post showing, but the seat is at the correct height and could be adjusted lower still. Since I am using touring bars (it came with north road and I put on albatross), the reach is not excessive at all because my bars come the completely opposite direction that drop bars do from the head. So in fact, unless I was using drop, bull or tri-bars, the sizing methods would suggest a frame with too little reach.
The frame-size finding methods are useful, but they are just a starting point. You can make many adjustments, though some will cost additional parts. Knowing what you want to do to the bike and with the bike in advance of selecting the size will help.