Hi Capnahab, spanners don't normally fit exactly snug on the size nut they are made for and there is always some amount of slack. The slackness varies from one manufacturer to another and ring spanners very often have a little tad more than open ended ones. A 13/16" AF spanner should fit a 9/16" unified (UNF & UNC) nut/bolt, however it will fit a 1/2" BSW or BSF nut/bolt. On an old spanner, you may well see the spanner marked as 7/16" BSW/1/2"BSF as shown in the photo below.

In this photo is a 1/2" whit bolt with a clearance for a 0.2mm feeler gauge, but in the next photo is the same nut in a 13/16" AF spanner but it will only accommodate a 0.05mm feeler gauge.

So if you can only get a similar feeler gauge or smaller or none at all in with you 13/16" AF spanner, then your nuts are most likely 1/2" BSW or BSF, you will need to find out the diameter of the bolt and the TPI.
It is also possible that your nuts are old enough to be from before Whitworth nuts/bolts were reduced down to the same size as BSF in which case they could be 7/16"BSW.
Regards Nick.
Edited By Nicholas Farr on 27/06/2015 21:00:36