It's hard to quantify if this was a "common" problem or not -- the sinkings of the Vasa and Mary Rose are separated by about 80 years, and the ship types were quite different -- but there were certainly other ships sunk by water coming in through lower gunports in a high sea, such as the Thesee at Quiberon Bay in 1759.

But the phenomenon does point to a common issue about ship design in the early modern period, which is that it was extremely difficult to predict a ship's buoyancy or stability without modeling, and that there wasn't any modeling (even static modeling in a basin) or even the maths available to calculate buoyancy in this period. Every ship that was drawn up and built during this period was the product of empirical thinking, which means that ship designs were inherently a bit conservative, and didn't change much over time. As a sailor, the ocean is always trying to kill you, so you tend to be very wary of innovation.

That said, there are some similarities and also some differences among the experience of the Mary Rose and Vasa. The obvious similarity is that they were both swamped by a sudden gust of wind allowing water to enter through the bottom gun ports, which points to a misunderstanding of the effective heel of the ships and their ability to recover from it.

But the ships were built very differently from one another -- the Mary Rose isn't shown here, but its profile is more like the Henry Grace au Dieu or even the Grace Dieu in this graphic than the Vasa is. Here's a detailed model of the Mary Rose, and one of the Vasa (click through the slideshow).

The difference in how they were built is related to how they were intended to be used. The Mary Rose is an older type of ship, and though the museum notes say it carried 60 guns, they would have been mostly small guns, not the large uniform armament of later ships of the line. It was primarily meant to pepper enemies before coming close and boarding, and it was modified to make the "castles" at the bow and stern of the ship taller, which contributed to its lack of stability.

The Vasa, on the other hand, was trending towards being a ship more like the line-of-battle ships that we think of in the 17th-19th century, that is designed mostly to be a heavy gun platform. Vasa was also built by man, Henrik Hybertsson, who does not seem to have built a two-gundeck ship before, and it has a hull form (with a shallow hold) that seems designed as though they simply inserted a second gundeck on top of the first one. On top of that (no pun intended), the ship was then covered with ornamental sculptures that made it more top-heavy. A rolling test of the ship (the ship's construction supervisor had men run back and forth from one side of the top deck to another) was so terrifying that the vice-admiral in charge of procurement stopped it; however, the king of Sweden was impatient to have the ship put to sea, so it went out into the harbor, was hit by a gust of wind, flooded and sank.

In both cases, what's technically the issue is that the ships' metacentric height was too low. The metacenter of a ship or any floating body is essentially the point around which it oscillates when it rolls -- this is a bit technical, but think of the metacenter as where the ship rotates. It's situated above the ship's center of buoyancy, and as a ship rolls there's an (imaginary) "righting arm" or "righting moment" that's produced as the center of buoyancy moves laterally with the roll. The basic idea is that the ship will roll, the wetted area of the hull increases on one side and will create a righting moment, and it will roll back the other way, etc. A very tall metacentric height means that ships roll quickly and quite uncomfortably, while a low one means they may roll and not recover from the roll, so the "high enough but not too high" sweet spot is difficult to hit. This is complicated further in the case of 16th/17th century ship design by the fact that there aren't internal subdivisions of the gundecks, so what's called the "flat surface effect" takes hold once water comes in. Think of carrying a big flat tray of water and trying to make it stable -- once the water sloshes to one side or another of the tray, there's a feedback loop where the center of gravity moves further in the direction of the sloshing movement. In the context of a ship, that makes it more difficult to recover from the roll.

Naval designers of this era didn't have the technical language to discuss this, but they understood the basic principles and certainly noticed how high out of the water a ship's gunports were when they launched one. Girdling a ship -- that is, adding extra timber at the waterline -- was common enough that even Sir John Narborough recommended it for his new first-rate Prince in 1670, saying in a letter "Girdling the ship would make it one of the finest ships in the whole universe, for it would make her much more floatier and carry her guns higher, and she would bear the better sail and be a better and securer ship to receive shot, and I believe it will not prejudice her sailing." Prince's lower gunports, as built, stood 3'5" clear of the water. (Quoted in Rodger, The Safeguard of the Sea, pp. 218).

In the long term, naval architects realized that they needed to do a few things to ensure stability. One was to make underwater hull forms deeper relative to the height of the ship (the Dutch still built flat- or shallow-bottomed ships, but they were constrained by their harbors). Another was to make ships wider relative to their overall length -- Vasa was 155 feet long between perpendiculars and only 38 feet wide, about a 4:1 ratio, while about 130 years later, the Victory was (is) 186 feet long, 50'10" wide, about a 3.6:1 ratio. That's not a radical change, but it does have stability effects.

Anyhow, this is getting long and perhaps rambly. I wrote more about ship design and construction here which may be of interest, and please let me know if you have follow up comments or questions.