MySQL Indexing and performance when size of index is greater than size of data in table
As you mentioned of 5.6 i guess tables are Innodb... Where we have PK stored as clustered index. The rows are stored in an index or say leaf node of the PK  index is data itself. 

Secondary further contains the PK key columns as well. So a long PK key => longer Secondary index (WRT size).

About large indexes, you might need to look out for duplicate indexes.
Also one thing to make sure is having large enough innodb-buffer-pool to make sure indexes are in memory for quicker access.

Refer:


https://dev.mysql.com/doc/refman/5.6/en/how-to-avoid-table-scan.html
https://dev.mysql.com/doc/refman/5.6/en/innodb-index-types.html
https://dev.mysql.com/doc/refman/5.6/en/mysql-indexes.html