The hedge-fund enclave of Greenwich, on the Connecticut Gold Coast, is about 100 miles and a world away from the capital.

But the fiscal crisis in Hartford, the historic center of the American insurance industry, is fast becoming more representative than mansions or yachts of the wealthiest state in the U.S. The city is edging closer than ever to the breaking point, waiting for the financially troubled state government to step in.

It may seem crazy that a place as rich as the Nutmeg State, which counts among its residents hedge-funds masters like Ray Dalio and Steven A. Cohen and legions of Wall Street bankers, could be in such fiscal trouble. Last year, the per-capita income there was $71,033, the highest in the nation, according to the U.S. Bureau of Economic Analysis.

For all that, state-worker pensions have been underfunded for decades. Tax increases aimed at closing deficits have put