Does a freezing point of a substance always increase as its boiling point increases?
Yes and no.

Yes in the sense that boiling point and freezing point usually correlate. If there is a line of homologous compounds, than this properties rise in general, as both are increased with increase in number of dispersion interactions involved. However, it is likely to find several compounds with odd behavior, especially around beginning of the line. 

As a counterexample. Toluene has higher boiling point than benzene, but much lower freezing point. This is because toluene has less symmetrical form and consequently has harder time packing into strictly ordered lattice. 

Another counterexample. Gallium melts around 30 celsium, but boils well above 2000 celsium. However, Tallium melts around 300 celsium, but boils around 1700.

However, for similar compounds the values usually correlate in the series. Say, for alkanes both values steadily grow with increase of number of carbons... give or take occasional peaks due to details in crystal packing.

However, if you consider compounds of different families, it is not predictable without looking into structures of the compounds.