Email quotas are something that is maintained by the system administrators. In order to be able to send bigger attachments, they will need to change their configuration.
The biggest reason to choose to not allow this, is that if you can send bigger emails, emailboxes grow significantly larger quickly. This means that email boxes themselves need larger quotas and thus servers need more harddisk space. More space means more costs in the end, but it depends on how futureproof your company is.
For example, if you have Office365 email, you get to deal with quotas of 50gig per mailbox user, but then, Office365 does not restrict in small attachment sizes either.
Secondly, enabling this for emails inside the company is one thing, but it doesn't matter what setting you change, as soon as the email leaves the company, it is also up to the receiver if they can actually receive emails with large attachments. Often for the same reason as above, a maximum email size of 5mb is permitted, anything above is automatically discarded and an email is returned stating the mail was not delivered because the attachment exceeded the maximum allowed.
If you are indeed sending emails to addresses outside of your company, it is very well possible the error you received was actually from the remote server. The way to know for sure, If outlook gives you a message box before hitting send, its on your end. If you receive an email with a NDR (Non-delivery Report), then its on the remote side.
Lets say the mail is indeed going to remain inside the company... From an IT perspective, if the file is going to be shared between someone and remains inside the company, why not place the file on a network share and in the email tell them where to find it? If it is not for everyone's eyes, then ask your IT department to create a folder for you that only you and said persons have access to, and explain you have to send files of a large size and that email is not working. They will gladly setup this folder for you.
So long story short, If the email stays inside the company, ask for a network share and explain why you need it. If the email does leave the company, you will have to find alternatives.
You mention OneDrive or other sharing sites. This is indeed one of the methods you can use. I personally would use www.wetransfer.com as service. This is a service specifically used for sharing large files over the internet with someone else. If you fear that the content can only be read by the recipient, then use a program like Winrar or 7zip, zip the file into a password protected archive, upload and send the file using wetransfer, and email the password separately. Then there is no way for anyone other than the recipient to access the content of the attachment.
