Yes, Google limits the amount of mail you can send over a 24 hour period. The restrictions limit how many you can send per day, as well as how many recipients per message.
Once this limit is reached you will be unable to send messages for the remainder of the 24 hour period. Google may also lock you out of your Gmail account. If this happens please contact the support desk to get your Google account unlocked.
Recommendations for sending bulk email
Here are some ways to avoid exceeding limits when sending a large amount of mail, such as for a marketing campaign:
Stagger mass communications over 2 days—Send messages to one group of recipients, wait 24 hours, and then send to another group.
Send mail to a group address—Send mail to many email addresses using a single group address. For example, contact all users in your organization by adding everyone in the domain to an email list. For more details, see Groups and Group policies and limits.
Check out MailChimp—MailChimp is an email marketing productivity app that offers bulk mailings. For for information see our write up.
Use the APU Mass Email Tool—If you are just sending messages to students (APU accounts). This could be the option for you. See the product description page for more information. Please note access to the the Mass Email Tool requires approval.
Review bulk sending guidelines and best practices—Some email recipients might report bulk messages as spam. Therefore, you need to ensure that all recipients are willing to receive the emails. You can also check email settings for any incorrectly configured forwarding rules or filters. For details, see Google's bulk sending guidelines and best practices.
SOURCE MATERIAL: https://support.google.com/a/answer/166852?hl=en