Cant get my rear derailleur to reach all cogs
As you suggested, the cable may be at fault.  I change my gear cables once or twice a year to keep shifting smooth and free. If the cable/cable housing are old and worn they could be offering resistance.  If that resistance is too high the springs in the derailleur may not be strong enough to pull the cable through.  

Try removing the cable and housing, then make sure the derailleur and shifter are moving freely before fitting the new cable.

Then replace the cable and housing (these are cheap to buy online, but you do need a pair of housing cutters to cut the cable housing - think mine cost be about £15).