In short, yes.
All carriers allow messages containing more than 160 characters but each 160-character segment is recognized as a single message. The difference is in how the message will be displayed on the recipients' phone rather than by the carrier.
Smartphones (e.g., iPhones or Android phones) will reassemble multiple 160-character segments into a single message in the correct order and the message will appear as the sender intended. Carriers do not always send messages in the same order they are submitted so standard phones ("flip phones" or otherwise called "feature phones") will receive the message in multiple segments and may not display them in the correct order depending on how the carrier sends them.
Regardless of which type of phone the message(s) are sent to, the sender (you) will be charged for each 160-character-or-less segment.