Well I started playing about 2.5 years ago with my only musical background being a basic understanding of how to read music. If you can read music, that makes the start up so much easier. I haven't done any exams yet, but over the course of the years I've played the ABRSM grade pieces for grades 1-5. Currently somewhat comfortable with the grade 5 exam pieces now so it can be done. Grade 5 pieces are comfortable, and I can make a stretch to pieces just above that level (e.g. Chopin Op 69 no 2, Mendelssohn Op 19 no 6).
A HUGE part of it really is how much time you're willing to put in. I needed 1.5-2hrs daily. The longest I've gone with no piano contact at all in the past 2.5 yrs has probably been less than a month. All I can talk about is playing pieces at a grade 5 level. I know right now I'd barely pass an exam because my sight-reading is awful and I haven't done much work with sight-singing. You'd need to also work on those practical musicianship aspects also to pass the exam successfully. But for the older (and more importantly serious, dedicated and patient) beginner, it certainly is quite possible to reach grade 5 in that time frame. Actually, some might say it's expected, but everyone's different. Just take your time, get a good teacher and get prepared to work!