I have an existing used Schengen visa from the 21-July-2015 to 20-July-2016 issued by Belgium. I recently applied for a Schengen visa from Germany & was issued a visa from the 19-July-2016 to 18-July-2017. As you can see, there's an overlap of 19th & 20th of June. Would this cause an issue when I'm traveling? The agents at VFS assured me that this wouldn't be an issue but can someone please confirm?
As @HenningMakholm mentions the belief that it's illegal or impossible to have two overlapping visas is misleading. The Visa Code (EU's main law on the Schengen area) mentions nothing about overlapping visas. However the Handbook for the processing of visa applications contains the following recommendation:
A holder of a multiple-entry visa may apply for a new visa before the expiry of the validity of the visa currently held. However, the validity of the new visa must complement the current visa, i.e. a person cannot hold two uniform visas valid for the same period in time.
Note that the rule is for the consulate, not for the applicant. Hence you cannot be held responsible for the consulate's mistake in issuing the visa correctly. Seeing that the previous visa has already expired you shouldn't have any problems at the border.