Other than cost, is there a reason to not use higher rated components than called for by design?
Yes, this is valid and commonly done for one-offs produced from local "lab" stock.  There is cost in maintain more parts, which can easily dwarf the savings of using 5% resistors instead of 1% resistors in those cases where 5% is good enough.

There are also costs in production for each different part used.  Even at high volumes, the pick and place machine has to be set up separately, different reels need to be bought, kept somewhere, etc.  Unless you have a very high volume product, it makes no sense to use a 10 k&Omega; 5% resistor in one place when you need to use 1% tolerance of otherwise the same resistor elsewhere on the same board.

In other cases you have to be careful that the better spec in one dimension doesn't cause tradeoffs you care about in another dimension.  For example, you mention higher voltage capacitors substituted where lower voltage is required.  That's OK electrically, but higher voltage caps will be physically larger most of the time.  The same is true for higher wattage resistors.  Electrically, a 2 W 100 &Omega; resistor is a superset of a &frac14; W 100 &Omega; resistor, but the 2 W resistor will be significantly bigger, which may incur other costs.