The planets in our system are most often grouped into two categories:
Terrestrial:
Gas Giants:
Jupiter
Saturn
Uranus
Neptune
Why is it that we don't see "in-between" planets in our system - large, rocky worlds larger than Earth but with thick atmospheres somewhat similar to those of gas giants? Are they possible? Did they not occur by random chance?
Super-Earths and Mini-Neptunes are the "in-between" types of exoplanets you're looking for. A sweeping generalization would put most in the range of $\sim1$-$10M_{\oplus}$ (Earth masses), with some outliers a bit above that. They may have significant quantities of hydrogen and helium in their atmospheres, as well as water, in liquid or vapor form. The latter should also have what are referred to as volatiles (colloquially called "ices") - compounds such as ammonia and water that are present in the atmospheres of proper ice giants like Uranus and Neptune.
One reason we don't have any of these in the Solar System is simply that no worlds with the proper mass formed. I explained here that there is a transition region of sorts, but most super-Earths require minimum masses of $1.5$-$2M_{\oplus}$ to keep their large hydrogen/helium envelopes (which terrestrial planets may accrete early in their lives but later lose). There would have to be a planet just in that mass range and at the right distance from the Sun (see D'Angelo & Bodenheimer (2016) for general constraints on semi-major axes). The environment in the protoplanetary disk may simply not have been suitable to produce the right body at the right time.
This is not a characteristic of the solar system. It is a characteristic of the definitions of the names you used. Neptune and Uranus are the bodies you believe to be missing. In fact, with the mass of Earth at 6*10^24 kg, Uranus at 9*10^25 kg and Jupiter 2*10^27 kg, you'll notice that Uranus is only ~15 times the mass of Earth while Jupiter is ~20 times the mass of Uranus.
The initial growth rate of planets was limited by how much solid dust there was that could clump together to form them. Planets that formed in colder conditions outside the ice line were able to grow faster because of ices (especially water but also ammonia, etc. depending on how far out) that were prevalent in those parts of the nebula from which the solar system formed. Once they reached a critical mass and were able to hold on to hydrogen and helium gas gravitationally, they would grow much faster.
Jupiter grew fast due to its location just outside the ice line, where it had a comparatively dense supply of ice, and could then quickly gobble up a large supply of gas. Neptune and Uranus did not grow as quickly and had only collected a small amount of gas by the time the solar wind blew the nebula away. This is why they are composed of a much larger percentage ices than Jupiter and Saturn; in fact they are often called the "ice giants" which is the correct intermediate classification. (The "ice" in "ice giant" refers to the substance, which is of course no longer generally solid in the atmospheres of these two bodies.)
Since 2016 a ninth planet is hypothesized beyond the Kuiper belt which, if it exists and if it's a planet, is either a super-Earth or a mini-Neptune.
But you might want to consider Venus a planet in-between. While it isn't bigger than Earth, on the surface of Venus the air pressure is almost a hundred times that on Earth which makes the atmosphere a supercritical fluid there like those of the gas giants. For comparison, the air pressure within Uranus and Neptune is more than a thousand times that of Earth and within Jupiter and Saturn more than a million. So Venus is kind of inbetween. From space, Venus looks similar to the gas planets since you can't see its surface at all, but its cloud layers.
