Is it true that Google cuts you off after sending a certain amount of emails?

Connect/Google Apps does limit the amount of mail a user can send. You can review the limits here, which restrict the number of both messages and recipients per message. After reaching one of these limits, you could get an error message and your account might be suspended for up to 24 hours (you won’t be able to send any mail). If you need to send more mail than these limits allow, please contact your Tier 1 support personnel or open a Help Ticket for information on alternate options.