Check the file size. Uploads are limited to 200Mb for images/files and 1GB for video. If your media is larger than this, you'll need to compress it further, or (if it's a video) host it off-site and stream it into a Video Element.
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Check the file format for video. Videos need to be .mp4. If you upload a .mov file it will not play since it's not a web friendly format.

Browser extensions and add-ons. Try temporarily disabling any browser extensions or add-ons, then try uploading the files again. If you can add files now, that means the extensions are the culprit.

Firewalls. Try temporarily disabling any software or hardware firewalls installed for your computer or your router.

Proxy server. Proxy servers are an intermediary between your computer and the servers that host TIME Sites media uploads. If you use a proxy server, try accessing your site from another location or internet connection to see if that resolves your upload issue.

Wireless Internet. If you're using a wireless Internet connection, try plugging directly into your router or cable modem. Wireless connections can sometimes be unstable or inconsistent.