SP4T slider switch on two states at the same time
Is this a sign of a bad switch? 


Not necessarily. Switches come in "make before break" and "break before make" variations. 

In some applications it might be desirable to always have the common connected to something rather then let it float. e.g. When switching between two audio sources it may be better to short them together rather than have the audio floating during transition and picking up a hum. Use "make before break".

In other applications it may be undesirable to have a short. Use "break before make".

Even if it is, are there any way to prevent two LEDs lighting up at the same time with this switch?

Not easily. You haven't shown your circuit. There may be some means of doing it.



Figure 2. A typical low-cost 4-position switch.

If the switch is of the type shown above you should be able to open it by unfolding the metal tabs underneath. Examine the contacts and see if you can file away enough between them or modify the wiper contact carefully to change it 
into a break before make. Obviously, this would be best attempted on a spare switch.