ATT's CTO recently said that it's not just the iPhone that's caused the overload. He says the 18x increase in data over the past few years is also caused by a huge customer uptake in phones with quick messaging built-in. He also says the number of ATT voice calls doubled in the same time.
It follows that all carriers have had increased voice and data usage for similar reasons. We know that the number of people who've dropped landlines for cell phones has increased incredibly over the past years, and now there are more homes with just cells than just landlines.
So it's not an iPhone specific problem to begin with. However, as whether other networks would have the same struggles, the CDMA carriers would have very different symptoms.
Verizon keeps 3G voice and data communications separate. This means that an overload of 3G data users could not cause dropped voice calls, as happens with ATT.
Verizon and Sprint also designed their cell network for CDMA radio coverage from the start. This is critical planning. ATT did not, since they started with GSM, and this shows up with poor 3G WCDMA cell breathing behavior.
So while they could have similar data overload throughput problems, the dropped call and/or fallback to 2G problem would not exist in anywhere near the same way.
