Depending on the complexity of what you do, yes you can do both. However, people who do both tend to concentrate on the programming end and put off the dba tasks because they generally come from the programmer world not the dba world.
Access has basically no tools for performaing database adminstration, so you can get away with this. But SQL server will require you to learn more about database administration and performance tuning if you want it to perform well as your database grows.
It is considerably less difficult to be a dba for SQL Server than for an Oracle system which often requires multiple dbas. Most tasks can be set up to run on a schedule or when an event occurs, so you may not have to spend much time doing dba work on a day-to-day basis once you have things set up.
I do agree that the best programmers understand database concepts, design and basic dba tasks and the best dbas have an understanding of what the programmers go through.
However, there are distinct differnces as the dbas are more concerned with the day-to-day operation of production systems and programmers are generally concerned more with development and the maintence of code.
A large shop might have dbas, database programmers and application programmers as well as systems analysts, architects, testers. etc. The smaller the shop, the more hats you wear. I'm a dba and database programmer, but I have done some .Net web pages in a crunch and I have written requirements, developed test plans, written sales proposals and a whole lot of other things because I work in a small shop.