Technically yes, but FiOS's internet service is overprovisioned for that very reason. You'll still obtain full advertised speeds even when you have multiple VOD streams playing. When you order 50/35, you actually get ~60/40, and so on.
If you have 6 boxes all using VOD at the same time, you may see some hit. VZ used to dynamically adjust the cap to compensate, but I dont think they do that anymore?
The biggest issue with cable's approach is that Internet, phone, VOD, and conventional TV all share the same coax. Each channel is around 6MHz and most cable companies use either ~750 or ~815MHz. This limits how many channels they can dedicate for traditional TV, internet (cable companies now use up to 12 channels between downstream and upstream for internet alone), VOD, etc.
Verizon has plenty of "space" available on the fiber, so they split it up there instead of in the Coax RF spectrum.
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