Bob's solicitor could not have sent the letter to himself, could he?
This is a nice observation, Sherlock! (a joke)
I suppose it could be done but it is probably rare

. I sent a letter to myself once when I had written a song that I wanted to protect from being stolen from me. I sealed it in an envelop and mailed to my home address. When I received the envelop I did not open it but placed it in my safety deposit box for safe keeping. If someone tried to claim the lyrics as their own, I could sue him/her at the appropriate time. I would present the unopened, time stamped envelop and its dated contents to a judge showing that I had indeed written the song and the lyrics were stolen. It was a cheap way of protecting my creation. I suppose the song is not that good since no one has seen fit to steal the lyrics and the envelop still lies in the box.
So in summary there may be a legal reason for a solicitor to mail himself a letter, but I don't see why Bob would be worried about it.
Bob asked the post office to find out why his solicitor's letter was not delivered to him.
I agree with @andrewg927. There is nothing with your original sentence and I don't see why there should be any confusion as to who "him" refers to. It is obvious to me that "him" refers to "Bob. I also don't believe you should change "solicitor's letter" to "the letter sent to him by his solicitor". This is not an improvement on your sentence at all.