I just wanted to point out a few things about the iphone as this is the phone we are familiar with. We also use AT&T.
...I can tell you absolutely that you need to turn your phone completely off. Airplane mode is not 'turned off' and you MAY still incur roaming charges. ... So, best bet is to simply turn it off completely and leave it be!
Airplane mode will prevent the phone from sending or receiving calls and data. The problem as another poster pointed out is that the phone can accidentally be put back into regular mode. It is when this happens that charges may occur as the phone is searching for a network. Being a smartphone it uses a good amount of data doing this.
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The Iphone is never really, ever, totally "off". Even if you turn it "off" it is still communicating with GPS satellites and cell towers, etc. This is why you can do things like "MAP MY RUN"--you start the run timer, then turn the iphone off, then turn it back on to end the route and check your time, etc.
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To avoid any ROAMING charges on the Iphone 4, put the phone on AIRPLANE MODE.
To be extra safe, we went to Settings--General--Network and turned "cellular data" OFF. Didn't want to take any chances!
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The iphone actually can be turned off, if you power it off, you hold the power button down until the slider comes up which says "slide to power off". When it is powered off and not just in standby there are no signals that you are charged for coming to or from the phone. I agree about the airplane mode as noted above.
I also agree about the turning cellular data off, if you are going to use airplane mode and are worried that you may accidentally turn airplane mode off, then yes this is an excellent idea. Turn off the location services and turn off cellular data. If you have a previous version of iphone os you can also turn off 3g separately.
To avoid any accidental charges on our cruise, we took the batteries out of our smart phones. I didn't want to take a chance of them getting bumped as we took other things in and out of the safe and coming back on and doing any kind of connecting. Plan to do the same thing in June.
Certainly taking the batteries out is the similar to turning the phone off. However you may be able to do this on most smartphones but the iphone is sealed.
We have been on several disney cruises where we kept our phones on and also we spent a few days at DLP. So I just wanted to point out a couple of things that we did.
We usually use our phones for music and videos so we use airplane mode for any phones that we are not planning to make phone calls with. As a precaution we turn off location services and cellular data before enabling airplane mode. We have never been charged for this in the past 4 cruises.
If we use DH's phone for calls we do the same, turn off location services and cellular data. The phone can still get a signal for calls and regular texts but the multimedia texting is off (can't text photos or videos), we don't' check email or use the internet (obviously as we turned data off). We simply use the phone to send and receive phone calls and texts. How much does this cost? Well it depends on where you use the phone. But say for example you are just using it on the cruise ship you will be using the Cellular at Sea Service unless you are near a port. The cost is $2.49 for calls received and placed (missed calls don't normally count, but remember no visual voicemail if you turn data off), Text Messages are sent at $0.50, received texts are billed based upon the plan you have at home. So the cost is not that bad. You actually don't even need the international cellular plan or the international text plan since cellular at sea is excluded from these programs.
If your scenarios changes then you have to base your pricing for data and voice on the country you are in, here is the att site to give you an idea of those costs
HERE.
I think that there are just a lot of accidents and misconceptions about actually using an iphone or similar smartphone.
- Turning the phone off - this means actually powering it off or taking the battery out.
- Standby - people off think this is off because the screen is off, but the phone is on just waiting for you.
- Airplane mode - this mode disables the voice and data signal as well as wifi - no charges for this mode, and interesting you still have to turn it off even though this is an airplane mode.
Like I said before we have done this on several cruises and on our trip to Paris and Barcelona. We usually count on a few phone calls during the trip, the most we have spent is about $50 above our regular phone bill. If calls are not your concern then airplane mode with caution or powering your phone off will save you the trouble.
I have had to do this a few times to reset it after it has frozen (before trading for a new one, my first one had issues) The tech support guy told me to hold it for several seconds after it turns off, like you said. It works wonders for when it freezes or is acting crazy! 
