My apologies - I am talking from experience, but in relation to one part of the country. Maybe I was a bit presumptuous in extrapolating my experience to the whole of the country.
In Vinales, Pinar Del Rio province, a separate internet cafe was opened for cubans some time ago when demand from the locals became so great. Cubans could use this cafe for email access (I’m not sure about web access) at much reduced rates. This cafe has now been closed for some time, with no apparent reopening in sight. The other internet cafe for tourists continues to function normally. Technical reasons have apparently been cited for the closure of the cafe to cubans, but it seems rather odd that the closed internet cafe is only across the road from the tourist internet cafe, which has no apparent problems.
In relation to text messaging, many home phones that have been installed in the last few years are effectively mobile phones without batteries and so only work when plugged into a power supply. These phones have the capability of sending and receiving text messages, just like any other mobile phone. It was possible in the past to send an international text message to one of these phones (outgoing international text messaging was not possible). It does not appear to be possible anymore to receive an international text message on one of these phones.
Maybe someone could tell me from experience that this is only a local issue, not a national one.