Yes, it is possible to be both "progressive" and "conservative." This Gallup poll result indicates that 22% of self-identified "progressives" also consider themselves either "conservative" or "very conservative."
"Progressive" simply means that you want the place to become a better place. You want to do something to achieve that end. It doesn't necessarily mean that you believe liberal approaches are the way to do that. Generally it will--77% of progressives identify as liberal or moderate--but not always.
Personally, I consider myself a "progressive" but I also have my conservative side. For all people over the age of 18, I'm a libertarian. For people 18 years of age and younger, I believe in the totalitarian police state. Plus, as a Lutheran, I believe in the intractibility of sin and therefore have a pessimistic view of human nature, very conservative that.
On the other hand, if you want to make the place better, that means you want to do some things. Conservatives rarely want to do anything--in fact, the whole point of being a conservative is not to do anything. Liberals do want to do something, and liberals have programs, quite often very good ones, like Social Security.
Teddy Roosevelt, a Republican, once called himself a "modern progressive." As I recall, Hillary Clinton described herself the same way during the 2008 campaign, yet she's quite conservative when it comes to foreign policy. It means you want to make the place a better place--not necessarily more liberal, though that's usually how it tends to work out.