Users with multiple 32-bit Oracle client versions installed on their machines need to make certain that the path to the proper client is listed first in their PATH variable to successfully connect to the database. If there are other non-Esri programs on the machine that have this same requirement, the PATH variable must be changed to accommodate that software's requirements and then changed back to allow Esri software to successfully connect.
When a connection to an Oracle database is initiated with ArcMap, the Esri libraries attempt make a call to Oracle's Call Interface (OCI) function to create the connection by loading the Oracle Client library. The library it loads is the first client that it finds within the PATH variable. If the required client version is not the first one found, this can result in an error.
In situations where the PATH environmental variable must be changed temporarily to accommodate opening ArcMap, the following workaround can be implemented.
SET path=C:\Oracle\Ora11g\bin;C:\Program Files (x86)\ArcGIS\Desktop10.3\bin; START ArcMap.exe exit
Running this script temporarily sets the PATH environmental variable to point to the alternate Oracle client and open ArcMap. These environmental variables only persist for the ArcMap session and do not change the variables for any other programs being opened with their launchers. If another ArcMap session is opened without the batch file, the default PATH variables are used.