Cloud backup - ~3TB+ to backup

Greetings,

I am planning on using a cloud backup solution for off-site backup, but I'm concerned regarding the amount of data I need to store.  The initial upload is clearly going to be quite large - around 3TB starting.  I know Cox has bandwidth limits they traditionally don't enforce (including in my state) but was wondering if other users have had experience with this and encountered any issues/concerns.  Equally, if a Cox rep could advise as to whether I should throttle this backup over a couple months or if I could pay a small bump to guarantee I won't get a nasty gram (I've seen that happen to other subscribers).

Once the initial upload is done the rest should be a trickle in comparison over time.

EDIT:

To confirm, this backup will be through a 3rd party and not Cox - like Amazon, Crashplan, or another.