Most likely no.
Voyager downlink communication (via its radio link to NASA's Deep Space Network (DSN) is not continuous. You can check the contact schedule at this Voyager site. If everything looks fine during one DSN contact period, and then at the next contact period there's no signal at all, there are myriads of possible causes, ranging from failure of the radio system or the attitude control system that points the antenna to Earth, to running into something.
If it runs into something, that something doesn't need to be very large. Voyager 1 is receding at nearly 17 km/s, Voyager 2 at ~15.4 km/s, so a BB-sized ice particle could sever the connection to the RTG (the radioisotope battery that powers everything). If rocky or metallic, it could penetrate the multi-layer insulation covering the bus and blast a hole in the main propellant tank...there are multiple soft spots. In any case, all we would know is that when we tried again to contact the craft, we heard nothing.
If, on the other hand, something happened while we were in contact, the radio signal's strength profile would give some clues, though not necessarily definitive. Most failure modes would have the signal die off rather slowly, some faster, some slower. For instance, complete loss of attitude control would have it die off very slowly, with a well-determined profile as the direction to Earth drifts through the HGA's main beam, then into side lobes. Catastrophic rupture of the main propellant tank (very unlikely!) would have it die off in tens of milliseconds, but not between individual bits of data, at the current downlink rate of 160 bps.
There are very few scenarios that would interrupt downlink mid-bit, and collision with something big is one of them. But there is more than one fast-loss scenario. An example might be catastrophic failure of the X-band transmitter's traveling wave tube (TWT), something like breaking the lead to the internal helix coil, though I suppose they would try later to see if they get an S-band signal. A power glitch that smoked both the S-band and X-band TWTs might prevent that. Anyway, there are non-collision events that could interrupt comm mid-bit. They are unlikely, but then again, so is running into something big.
The net result: we wouldn't know for sure.