Why was caloric theory accepted despite observations that heat was produced by friction? Why does the caloric theory of heat only seem to have been seriously questioned after Count Rumford did experiments with cannon boring? Surely there were plenty of everyday examples prior to that of heat being produced by friction, such as a saw getting hot cutting down a tree, wood getting warm when being sanded, sharpening a blade on a stone or even just rubbing two sticks together to start a fire. Is it simply that nobody asked the obvious question before Rumford - namely, where is the caloric coming from in these cases?
 A: There were two different theories of heat fluid. The earlier one, phlogiston, was introduced by Stahl in 1703 (anticipated by Becher in 1667, who called it terra penguis), and the later one, caloric, put forth by Lavoisier in 1780-s. The theory was successful at explaining several types of physical and chemical phenomena, namely heat exchange, combustion, latent heat in phase transitions, and decalcination (formation of metals). In decalcination for example, adding phlogiston to "earths" (ores) explained their acquisition of metalic properties. This should not be surprising, modern theories reproduce phlogiston/caloric theories in the effective sense (similar to effective "particles" like phonons) for most phenomena they were used to explain. 
Mechanical theory was considered as an alternative (proposed e.g. by Bacon, Boyle, Hooke, Locke and Cavendish, as Rumford will later acknowledge), but seen as inferior, if not ruled out. Black's work on latent and specific heats in 1757-1764, and Wilcke's in 1772, was seen as decisive in preferring phlogiston over it. It was hard to explain mechanically where the latent heat went, why capacity for heat was not proportional to density, or how phase transitions took place. Black, on the other hand, naturally explained it all, latent heat was due to phlogiston's ability to combine chemically with matter, and gaseous state was a solution of a liquid in phlogiston. The standard explanation for heat coming from friction was that the stress of rubbing surfaces together forced phlogiston to be pushed out from between the atoms, and it appeared as heat. See Chemical Revolution chapter in Friedman's book.
Lavoisier's revision concerned the effects with addition of mass in combustion (burning) that he was able to measure, which contradicted the phlogistonic view of burning: according to Stahl phlogiston escaped a substance, whereas according to Lavoisier oxygen was added to it. But Lavoisier's oxygenation theory still required a heat fluid, which he renamed into caloric, and it was also needed to retain the Black's theory of latent heat. In the notes on Kirwan's Essay on Phlogiston (1784) Lavoisier writes:

"when a metal is heated to a certain temperature, and when its particles are separated from each other to a certain distance by heat, and their attraction to each other is sufficiently diminished, it becomes capable of decomposing vital air, from which it seizes the base, namely oxygen, and sets the other principle, namely the caloric, at liberty...  it is principally and almost entirely from this substance that the caloric and light are disengaged". 

One weakness of the theory was that the released heat would obey a material conservation law, not grow proportionally to mechanical work as predicted by the mechanical theory. Rumford tried to show in 1798 that under the caloric theory the supply of the caloric would be virtually unlimited in his experiments with grinding to hollow out the cannon barrels. But he was not considered credible by contemporary scientists (like Dalton, Leslie and Henry), his motives were not exactly scientific either, his claims about "unlimited" supply exaggerated, and reproducibility of his results apparently questionable. Besides, electricity was known to be generated by friction too, and at the time its status as a liquid was non-controversial. So Carnot in 1820-s was still working within the caloric theory in his study of steam engines, in particular using the caloric water wheel analogy. 
Adoption of wave optics in 1820-s was a second blow, it became harder to explain radiant heat as caloric particles streaming through space. But even more definitive Joule's experiments on the mechanical equivalent of heat in 1842 were originally coldly received. Only after Thomson learned of the energy conservation law formulated by Mayer in 1841, and paid closer attention to Joule's experiments in 1847, and after Clausius reinterpreted Carnot's theory mechanically in 1850 things came together for the caloric theory to be displaced in 1850-s. See Fowler's Teaching Heat: the Rise and Fall of the Caloric Theory.
A: The caloric theory had explanations for heating produced by friction. 
The caloric was supposed to be stored in every object. One of the theories (I don't know if it's the only one) was that the friction produced some tensions in the bodies and these resulted in the release of some of the caloric stored inside the bodies. As the caloric theory assumed that there is some conservation of the total amount in the universe, the conclusion may be heating by friction will work only until all the caloric stored is released. Rumford argued that is can go for very long time and repeatedly without any observable depletion of the caloric. 
Still, his observations did not bring the immediate dismissal of the caloric theory. Probably some people tried to explain it somehow.
