Had a strange problem with an iPhone 5S. When it was connected to our home wifi it could send iMessages to another iPhone on the same wireless network, but not to phones elsewhere.

Methodically tried to solve it by various means, but eventually discovered the phone needed a top-up to restore its data allowance. So does that mean that iPhones can only send iMessages by cellular data, even when they're connected to the internet by wifi? Or, given that it could still send messages within the home, is there another explanation?

Nope, not at all.

I assume you mean it doesn't need to use cellular data, in which case I wonder why it only worked after a top-up? The only explanation I can think of is that one of the other fixes we tried (eg switching iMessage off, restarting the phone and then switching it on again) took a long time to take effect.

iMessage doesn't require mobile data if there's a wifi connection. However, I've experienced a couple of wifi networks where iMessage just refused to play ball, even though I could send messages using WhatsApp. Possibly something to do with the way the network was configured - I don't know.

You could test it now, anyway. Enable Airplane Mode then switch wifi on. You will now have wifi but no mobile data connection. Try to send iMessages and see what happens.

Hmm, may there are some constraints on your WiFi network?

If so, it may block the connection which iMessage tries to establish to Apple's server. As a consequence, your iPhone has to choose an alternative way to make the connection - cellular data network.

I always send my girl friend iMessage when I'm in some college building with airplane mode being enabled but WiFi is on.

This all depends on what contact details your using to send the iMessage.
If your just relying on WiFi then you need to send via your contacts email address, whereas if your using a mobile number you need credit as it will use mobile data.

Yes. Some public hotspots have this kind of blocking rules in case iPhone automatically upload photos to Photo Stream or backup your iPhone to iCloud, both of them are taken place in the background and take whole uploading channel.

Not remotely true.

It doesn't matter.

Are you sure about that?
I recently gave my son an old iPhone 4 without an active sim while he stayed at his Nan's, which he could send an iMessage fine over WiFi as long as he didn't use a mobile number and just used mine or his sisters email addresses when selecting the contact.

I don't know the reason for that, but it honestly isn't the case that iMessages to email addresses send over wifi and those to phone numbers send using mobile data. You can send to either using either method.

Yes I'll try that.