What were the techniques available if someone were to break their leg or arm around 1550? Before the full saw solution, what could be done?
Here seems a better place than /r/AskDocs for this.
The basic medical techniques for a simple fracture of a bone in the leg or arm have not changed since the Iron Age. A 1600BC papyrus (http://en.wikipedia.org/wiki/Edwin_Smith_Papyrus) has the fundamental system down - you set the bone by traction, immobilize it, and wait.
The Hippocratic Corpus, which saw its first translated European printed edition in 1525, contains a full set of instructions, including splinting. In effect, the Hippocratic methodology remained more or less unchanged for about three thousand years until x-rays made the bone alignment easier, and anesthetics made the bone traction easier.
I previously wrote a comment on treatment for broken bones in pre-modern times, and I stand by it. I'll just add a few points on to that previous answer.
First, to reiterate, treatment for broken bones (at of fracture of the limbs, anyways) have not really changed since at least the time when we started writing things down and probably not before then either given that we see fracture healing in prehistoric skeletal remains. The Smith Papyrus is acknowledged as generally the earliest treatise on the subject, and it's basic recommendations are the same principles current used for fractures, though with significantly less smearing the affected area with lard and honey. I note a few traditional practitioners of bone-setting and healing, and should note that certain parts of Europe likewise had these specialists, such as the medici ossium of Renaissance Italy.
An orthopedic surgeon I know once joked that setting a bone isn't medicine, it's carpentry. The point being that it's not so much the basic care of reduction and immobilization that changes over time, but the accompanying medicinal applications. So the ancient Egyptians went for honey and lard ointments, the Greeks and Romans basically poured wine on everything, ghee has a staggering amount of uses in traditional Indian medicine, there were various herbal poultices used throughout Mesoamerica, etc. Some of these applications would be localized to the particular ethno-regional tradition. So for instance, Cameron in (1993) Anglo-Saxon Medicine quotes a text from that time and people for splinting a broken limb and then applying a salve made from linseed and boiled elm. My previous answer notes that in Medieval England egg white and comfey would be applied. These sort of medicinal ointments and applications would vary across regions, times, and even practitioners.