Why don't we have in-between planets? The planets in our system are most often grouped into two categories:  

Terrestrial:


Mercury
Venus
Earth
Mars


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 &amp; 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.