60 volts is half of 120. It is almost certainly a "phantom" or "stray" voltage that results from harmless electromagnetic coupling from both hot and neutral (or ground), and it indicates that the metal you tested is not wired to anything.
The bad news is that your other test results could be the same situation, depending on the particular coupling at each box — if you read 120 volts between hot and box metal, you don't know whether the box is grounded or just happens to be coupled mostly to a neutral wire. A regular voltmeter, with a single test, cannot tell you whether there is a wire connection or just a weak coupling.
If you have a voltage tester of the sort that is not powered by batteries but has a lamp in it powered by the line, that can help, because it tests whether it is possible to draw power from the circuit, which can confirm a connection. Another tool is a multimeter with a "Low-Z" setting, which basically puts a resistor across the probes (thus drawing some current, if possible) and measures the voltage, giving you the same kind of answer as the lamp would but with an actual voltage reading instead of just glowing-or-not.
However, these tools will read zero for both "both probes on the same voltage" and "not connected". In order to confirm that something is connected, you need to make two tests, comparing the metal you're testing to two different other conductors (e.g. hot and neutral), and observe that one of the readings is zero and the other is 120 V.
There are better tests for integrity of ground, but they require more specialized equipment and safety considerations.