Hmmm, I think the short answer is that its possible, but the long answer is a bit more complicated.
I'm 100% certain that it did not go down the way Donald Trump said in his tweet. I say that for a couple reasons. First Twitter doesn't really allow its users to make long, nuanced political statements, so we shouldn't do what a lot of news outlets do and take it exactly. Also, Donald Trump has a tendency to say statements in the moment that are either wrong or ambiguous but do have a kernel of truth at their core. I believe that we're getting his stream of conscious thoughts. We haven't gotten these statements from other presidents because historically, the statements have gone through many layers of approval. Is that a good idea? euhhh, I would say maybe not, although he's president of the US and I'm not. So maybe he's on to something.
More likely is the fact that someone in Trump tower was under suspicion and being monitored by the FBI,CIA,NSA,etc. It's also important to realize how large and general our surveillance programs are already. Regardless of whether you have a warrant or not, anytime an international phone call is made, that is monitored by the NSA. Anytime data on the internet is shuffled to a server whose IP address is outside the country, that data is monitored by the NSA. This is public knowledge. The extent could be much larger, or it could have since shrunk. It sounds like someone in the Trump organization was making phone calls outside the United States frequently, and with the news about Putin messing with the election, intelligence agencies wanted to see if there was any merit.
We also need to consider a more Machiavellian scenario. I think very rarely are rules followed exactly 100% of the time. For anyone whose ever worked in a restaurant, think all of the times the restaurant wasn't in 100% compliance of the health code. Look at all miscarriages of justice that happen today with overzealous prosecutors who are so driven to convict someone that the wrong person is convicted, these things happen.
What does this have to do with Trump? We have no idea what oversight or checks are on the FISA court. We know that since 9/11, legislators have drastically the requirement of law enforcement agencies to search and wiretap, strictly for the purposes of gather intelligence to stop attacks like 9/11 but not to arrest or convict (the idea being if the guy on is just talking on the phone about going on a jihad you can't really arrest him for conspiracy to commit terrorism, but if he says hey billy were go do it at this time on this date and your arrest him in his car with a bomb in his trunk).
It's possible that rules aren't always followed in Washington. That reality of Washington today might be more like House of Cards than School House Rock. It's possible that FISA intelligence warrants are rubber stamped by the courts (the criminal search warrant was denied by the court but the intelligence warrant was approved for trump tower) . It's possible that some machiavellian insider wanted to gain info on trump but couldn't go directly, but instead needed some plausible deniability. Do I sound like a conspiracy theorist? Yup, there's definitely a little bit in this scenario. Do I think that this is what happened. No probably not.
However, let's be honest here: was Russia's intention when they hacked the DNC's servers really to make Hillary Clinton lose and Trump win? EVERYONE, the pollsters, the pundits,the news, and republican party members thought they were going to lose. Hell I bet even Mike Pence thought they were going to lose after that audio tape came about trump came out. If anything thing Putin hacked the DNC to send a clear and intimidating message to Hillary Clinton, the surefire winner.
I believe that this is the reason why Russia hacked the DNC. I can't tell you how said it makes me when democrats spin this to get ahead. I have a lot more respect for democrats who are attacking trump for what he's said about women then saying the US election was rigged by Russia because according to Buzzfeed, Russia has pictures of him peeing on prostitutes. Challenging elections are extremely dangerous. There was this one election in particular where the country was extremely divided. The candidate who won the electoral college only won 39.5% of the popular vote. This prompted half the country to consider the election illegitimate and prompted a civil war which killed 2% of the population and has divided the country to this day. I honestly believe that if the civil war did not happen, that slavery would have ended similar to the way it ended in Great Britain and many the problems such was the KKK, Sharecropping, Segregation, Jim Crow, either wouldn't have happened, would have ended sooner, or would have been less pronounced. Anyways, that why I guess you spend so much time on the Civil War in school because even in unrelated topics it has relevance.
Anyways, that's my wall of text, like to know what everyone else thinks.