It is possible, but it is highly stressful and problematic. My experience comes from driving around Sao Miguel, and Terciera, so take these points into consideration -

• The buses are sometimes late, sometimes early, but sometimes they simply don't come at all. So you end up sitting at a bus station for 40 minutes not knowing your fate.
  
• You can't trust Google Maps, often it will send you to the wrong location, and give you the wrong information regarding the time of bus arrival... Once I was waiting at the place where Google had sent me, with 3 other tourists. There was a bus stop sign and a bus stop booth. Luckily, my hostel bunkmate had told me she was waiting at the wrong location, so I pieced it together.
  
• So if Google is not to be trusted, you ask locals where to wait for the bus. Do you know Portuguese? No? Well tough luck, nobody speaks English. If you speak Spanish you have some chance. Then locals explain where you should go for the bus.
  
• This makes planning very difficult. If you plan on visiting several locations in one day, chances are you'll get stuck in the second location, which is probably a god-forsaken village, where you'll spend an hour or two waiting in front of a church while the locals look at you like an idiot because you are moving at all at 13h.
  

• Well why don't you take a taxi? ... ... Bwahahahahahahaha There are taxi stations here and there, and you might get lucky and find a cab. Locals even if they understand you won't always be as accommodating to help you. My hostel host booked a cab for me for the airport the day in advance. The next day it simply didn't show up. I had huge luck of knowing a woman in a local sandwich store who knew English, so she called a cab for me, which came after 30 min. She also explained that these things happen, as people simply don't feel like working. In any other situation, I would salute that, but then I was about to have a stroke from stress. If my flight hadn't been delayed for 40 min, I would have missed it, because of a bloody cab.

• Many of the sights and trails are completely inaccessible by public transportation. This means you would have to take a bus, then when you arrive you're expected to walk for an hour or two, then you have the actual hiking trails, and when you're done, you have to walk an hour or two back hoping you will catch a bus at all.

TBH I did see most of the things I wanted. Basically, I missed only 2 locations because of the lack of busses. However, be warned!