Can I recover old imessages that were deleted?

The only way to retrieve deleted messages is through a backup that you may have created on your computer or iCloud. You will need to restore your iPhone from the backup and it will restore your messages that are backed up. Note that should you choose to do that then the messages will only be as current as the day and time the backup was made. Check the backup date and time on iCloud and the backup date and time on your computer and choose the appropriate one to restore from.

Axel F.

Just to add to what Axel Foley has mentioned. If you restore from old backup you may loose everything else...

• If your backup is on iTunes: What ever you have created between the backup date and current date will be gone including Photos, Videos, Voice Memos, Notes, App Data, Health Data etc..
• If your backup is on iCloud: iCloud backup doesn’t include information already stored in iCloud, like Contacts, Calendars, Bookmarks, Mail, Notes, Voice Memos, shared photos, iCloud Photos*, Health data, call history, and files you store in iCloud Drive. Everything else will be gone Between the backup date and the date current date.