If you are out of memory on your iPhone, things won't work properly and yes, your phone can stop working. Open the Settings App. Tap General. Tap iPhone Storage. How much memory do you have available?
If none or very little, you'll need to remove things from your phone you don't need, which can include photos, music, apps, files, messages, email, etc.
Or maybe it's time to consider retiring your 10 year old iPhone, which can't be updated to any of the more current iOS versions and buy a newer, more modern iPhone with more storage as the phone you own likely doesn't meet your storage needs.