Yes, Bitcoin could theoretically run with any type of communication system, even physical mail, with everyone calculating blocks via slipsticks. However, such systems would greatly compromise network security and trust. Probably to a point that Bitcoin itself would be moot.
Have fun manually recording 50 transactions per minute and keeping an up to date ledger with 200,000 addresses...by mail. The window for double spends would be 4-7 days.
