Refer to [SIP Codes 603, 607, and 608] article for full descriptions of the response codes.

Short answer:

SIP Code 603 only indicates that a call was not completed without providing a specific reason why. 

SIP Code 607 indicates that the called party rejected your call.

SIP Code 608 indicates that an "upstream intermediary" rejected the call.

In each case, all AOI can do is attempt to work with upstream carriers to determine where the response originates.  Once that is done, it is up to the  customer to get their DID cleared.  

Note that trying to get more info from an upstream carrier regarding anything beyond the direct connection (IE, end users / equipment past Bandwidth, Intelliquent, etc) isn't worth the time as they will not have any means to obtain or verify those details.

The following website includes links used to clear DIDs from "spam" lists:  https://www.ustelecom.org/the-industry-traceback-group-itg/call-labeling-and-blocking-points-of-contact/