Let's say that a house connected to the power company in the USA has no ground rods (or pipes, plate, etc.) bonded to the neutral bar. Things should still work correctly because the current will be running from either hot wire (each hot should be at 120V) to the neutral wire, all contained within the lines from the power company transformer to the house. Even the ground wires at each outlet will work correctly because an accident from a grounded metal case will still trip the breaker at the panel.
However, this would be a "floating ground" situation. An issue with a floating ground, as I understand it, is that something can happen such that the voltage on the neutral wire is not actually 0V. For example, if you measure the voltage from an outlet in the house, it could be 480V at the hot wire and 360V at the neutral wire, which is still a 120V difference but maybe not what's connected to the outlet is expecting. I imagine this can be very dangerous because the ground and neutral wires are bonded at the panel, meaning a metal case to some appliance might be at 360V. Also, maybe there are some subtleties to how some appliances work and they cannot handle that magnitude of voltage. I do not know what other "bad" thing can happen without ground rods, short of a lightning strike.
So, what other "bad" things can happen without a bonded neutral? Also, in the above example where the voltages are incorrect, how can that even happen? If the power company has an issue with the voltages from the transformer, how does bonding your neutral to ground rods fix that?