Telstra, using Ericsson infrastructure, has demonstrated 1 Gbps Long Term Evolution 4G speed on a commercial mobile network.
Bonding of LTE channels (carrier aggregation) was key, allowing Telstra to use 100 MHz of spectrum across five separate 4G channels.
That is among the best arguments for allocating additional spectrum for mobile service providers, all other considerations notwithstanding. Use of more-efficient radios and small cell architectures can help maximize use of any specific amount of mobile spectrum.
Still, higher data rates require more spectrum, all those other techniques being used. To get more bits, you need more Hertz.