Actually, Electric Feild or Magnetic Fields do not cancel each other just because their field lines are in opposite direction. Fields lines are just representations of fields.
Electric Field or Magnetic Field as said earlier are vectors and can be added or subtracted.
Lookind alternatively at the definition, Field is a qty definrd as the force acting per unit charge (for electric field). Consider a situation that a charge is placed between two infinite planes charged with same polarity. The electric forces are equal and opposite and thus the net force is zero. Thus no force acts on the charge in that rregion so the electric field over there is zero. Same thing applies for magnetic field.
Hope u are clear.......
"If you win, you shall not have to explain and if you lose, you wont be there to explain"
~ Adolph Hitler