If I have a chain shirt with spell resistance 19 (+5 bonus), does that +5 apply to armor class along with the +4 from the chain shirt?
If you are referring to the table, Armor Special Abilities, please note, the "+5 bonus" to the right of "Spell resistance(19)" is under the column heading, "Base Price Modifier". This is merely informing you that spell resistance(19) has the equivalent cost or value of a +5 enhancement. Thus if the armor was a +4 chain shirt with spell resistance(19), the total value of the magical attributes would cost the equivalent of a +9 enhancement bonus to armor, or 81,000 gold pieces, but it would only grant a +4 enhancement bonus to the armor's armor bonus.
To be complete, the armor bonus is the total of the armor's armor bonus plus the enhancement bonus to the armor bonus (sounds odd, but this way they stack). Thus for a chain shirt, the item's armor bonus is 4, and you add any enhancement bonus to the armor bonus, so a +4 enhancement would make it a total of +8 to the wearer's armor class. Shields work the same way; the enhancement bonus is added to the shield bonus. Thus you can only have one armor bonus and one enhancement bonus to your armor bonus. The same is true of shields, you may only have one shield bonus and one enhancement bonus to your shield bonus.
Also, as mxyzplk points out, from magic armor
"A suit of armor with a special ability must also have at least a +1 enhancement bonus."
So even though the abilities have a cost equivalent to an enhancement bonus, the item must have an actual enhancement bonus to also have a special ability.