Sure you can. I just turned 50 (egad - how did that happen?) and have been seeing a steady increase in performance every year, since I started serious road biking about four years ago.
One of my benchmarks is a 25 mile loop over rolling hills. The day I broke 20mph average on that loop is a day I celebrated. Took me three years to do it, and I really had to fight off pain to keep that average speed up, but I did it.
However, you do need to push yourself a bit if you want to see improvement. Watch that average speed over a given route, and try to bump it up on each run. Do the same route two days in a row, and try to better it on the second day - you usually can. Every so often, try to ride with a group that's a bit better than you, one that keeps a bit higher pace than you do. You'll push yourself a bit harder when you start falling off the back. I occasionally ride with three women, all in their early to mid 50's, and they can whip me roundly on distance. One thing I love about cycling, is seeing the improvement. I'm getting stronger, not weaker. When you get to our age, that's a wonderful thing.
My current challenge is top speed. There's a hill near my house that has a brief steep drop, then flattens out into a very slight downhill grade. I'm trying to break 50mph down that hill. Two years ago, I hit 42mph. Last year, with the aid of a set of Zipp 404 wheels, I hit 47.8. I can make 50, it's just a matter of determination. If I don't get wiped out in the process... calls for the utmost in bike maintenance.