Was happening to my first S3 which I upgraded 2 months back.
After the problem (going to voicemail - SOMETIMES not ALWAYS) I replaced (Costco) for new one.
New one having the same problem !
Today I replaced SIM card also - no luck !
I am feeing bad about going to Costco again but I have only 2 weeks until Costco 3 month warranty period - so have to go again.
I am testing from my home phone to mobile. I am leaving voicemails for testing - that too are received very late.
The phone doesn't directly go into voicemail, but only after the ring (ofcourse no ring on S3).
Mobile network type - when incoming went to voicemail - LTE:14
Mobile network type - when incoming worked - HPSA:11
(Edit - Above might not be accurate information - Tried to call on S3 when it was showing LTE:14 & received the call)