A standard SMS has a maximum of 160 characters. Messages that exceed 160 characters will result in a 'second' message, though the patient will still see this as one long message on their phone.
When a message is longer than 160 characters, this is referred to as a 'multi-part' message since it contains multiple standard-length messages (or 'parts'). Since each standard-length message (or 'part') contains 7 characters for invisible headers and footers (these denote which 'part' of the message is being sent, e.g. part 1 of 2), the total character limit per SMS becomes 153 characters per 'part'.
Below is a guide of how many messages are constituted by each character range up to 765 characters.
1 - 160 characters = 1 message
161 - 306 characters = 2 messages
307 - 459 characters = 3 messages
460 - 612 characters = 4 messages
613 - 765 characters = 5 messages
Note that these ranges are calculated using standard English characters. If Unicode or Non-GSM characters are used, the ranges are slightly different.