There is no automatic shut off for data when you reach your cap that I've ever heard of. Not on any platform. The only thing I can think of is to get alerts from Verizon for data usage and then manually turn off cellular date (and rely on wifi) when you're close to your cap. Remember; Verizon's data usage on the MyVerizon website is only an estimate. Maybe turn on your usage alert for 90% and then turn off cellular data.
I'm glad that you asked. We do offer a feature that will allow you to do this. It is called Family Base. This feature will allow you to put a hard stop on a limit that you set. Once you reach that limit, you may not be able to access data unless connected to WiFi until your bill cycle starts over again. It is $5.00 a month, and here is some additional information about this feature: http://vz.to/1yy9XIR
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It looks like you have a blackberry and i'm not super familiar with those as of lately. I know on my S5 if I look under settings>data usage that I can limit mobile data usage right from the device. I don't need family base.
This is good if it's to help you from not going over, but if it's for say a child then it would be just as easy for them to go in and unselect the limit and it would continue to use data without a cap. Just remember to use a buffer; data usage isn't reported instantly so it's possible that constant video streaming could cause you to go over your limit until the data you used was reported and marked on your account. For instance if you want it to cap at 3GB I would set it to 2.5GB and see if that's enough of a buffer.
This is a great idea, but it's a little shady charging folks that are trying to avoid paying overages. I know Verizon is a for-profit corporation, and that it doesn't say "Red Cross" anywhere in their logo, but providing this as a free service to users would be a much classier way to handle overage charges. And it would engender a lot more good will from loyal customers. Just my opinion.
You could download a app or VZW text to tell you when your over your internet. Then in your phone depends on what kind you can turn off Data Only so everything will work but the internet and just use wifi.