The US debt markets are a relatively recent addition to the financial markets - the US government did not issue debt until the early twentieth century.
When the US debt markets started trading, they took their cue from the already established US stock markets where prices were quoted in eighths.
The site How Stuff Works give the following description of this tradition:
These early stockbrokers looked to Europe for a model to build their system on and decided to base it on the system of Spain. This was largely due to the fact that the U.S. dollar's value had been based on the value of the Spanish real.
The real was the Spanish silver dollar and was divided into eight parts.
This strongly implies the following route to thirty-seconds, which I give as an "educated guess" without reference:
As market liquidity increased, pressure on the bid/offer spreads prompted brokers (buyers and sellers) to "meet half way", thus introducing sixteenths pricing. Continued increasing liquidity and a "meet half way" attitude finally introduced thirty-seconds pricing. (Each of these pricing refinements would have cut into the brokers profit margins, so their loss would have to have been made up by increased liquidity. Otherwise, they would have resisted the change.)
This would then be the pricing practise inherited by the debt markets when they started trading in the early 20th century. The later introduction of options trading and the "hyper-liquidity" of modern debt markets introduced the final refinement of sixty-fourths.
As late as 1997 some of the older stocks in the stock market were still quoting in 32nds. This ended with the introduction of the "Common Cents Stock Pricing Act". The debt markets however appear to have decided to stick with the historic pricing practice, perhaps because the nominal price of a treasury bond is $1000 making further fractional price increments cutting into the brokers spreads undesirable to the market makers.