In really simple terms, there are a couple of factors.
First, the total mass of the induction motor rotor bars must be sufficient to handle the heat generated by the high currents during a transient (start) condition.
Second, the number of induction motor rotor bars (and thus slots) must be slightly different from the number of stator slots to minimize both electromagnetic and acoustic noise.
Third, the number of induction motor rotor bars must be enough different from the number of stator slots that the rotor MUST turn (e.g. cannot "lock" into place because of insufficient torque production).
If a bar (or the bar/ring joint at the end) breaks completely, there will be a measurable torque disturbance and slightly higher thermal loading on the rotor winding during operation. During a transient (start) sequence, it will not develop quite as much torque. The more bars that are damaged, the higher the disturbance and lower the torque. In order to avoid torque, noise and harmonics problems, there are specific rules for the selection of number of induction motor stator slots and rotor slot combination.