Yes. They are widely accepted in courts too as are e-mails. The default position is normally if the person received the text doesn't agree with the content, they will respond denying it. Silence normally = acceptance.

i.e. I test you and say "Don't forget the $200 you owe me by Friday", then you would normally reply either "Ok" or "What are you talking about - I don't owe you any money. What do I owe you $200 for?"

No normal person who doesn't owe the funds would just shrug and ignore it. They would dispute it and promptly.

Keep the texts and e-mails. They count just as much as a registered letter delivered by mail in this day and age.