I'll give the short answers to both questions, and then I'll explain the context/situation and what happened. This is a multi-comment post, so see the response to my own first comment for more.

1. It did actually happen. And yes, it did lead into the buildup of forces in the Sinai by Egypt, Israel's counter-mobilization, the expulsion of UN peacekeepers from the Sinai, and the blockade of the Straits of Tiran, followed by the Israeli preemptive strike on Egypt.

2. We can't know for sure. Some have said it was intentional because they wanted a war, others claim it was intentional but they didn't want a war, just tensions, and still others have other explanations.

So let's launch into it!

Question 1: Did the intelligence start a spiral? Yes.

It's now very, very well known that the Soviets provided intelligence to Egypt, claiming an attack was imminent by Israel on the Syrian border. It's also very well known that the intelligence was false. In fact, many Egyptian leaders knew it was false.

Before the play-by-play, you have to understand the context. Throughout the 1950s and 1960s, there have been a variety of flare-ups. For one, the Straits of Tiran had been closed to Israel until 1956, when the Suez Crisis occurred. Following that crisis, Egypt decided to allow Israeli ships through the Straits, though Israel warned Egypt that if it decided to blockade that waterway again, it would regard that blockade as an act of war and respond accordingly (this was stated in a speech to the UN General Assembly by Golda Meir, later to become Israel's Prime Minister in 1969). From 1957 to 1967, Israeli ships largely went undisturbed through the Straits, connecting Israeli markets with other areas as the port of Eilat continued to grow and develop (Israel's only other major port would be Haifa, which meant a long trip for ships through the Mediterranean and around Africa to reach Asia).

Despite the opening of the Straits in 1956, however, there were a variety of other factors contributing to tensions. For one, the armistice lines set following the end of the 1948 war were unstable, and became more unstable by the day. There were constant skirmishes on the Syrian, Jordanian, and Egyptian borders. Many of them related to attempts by Israel and the Syrians and Jordanians to divert the Jordan River, a major water source that all sides wanted to control more clearly.

Still other skirmishes related to cross-border incursions by Palestinian groups. Some groups operated with the consent and funding of Arab states, while others evaded Arab authorities while carrying out attacks on civilian targets in Israel. Fatah, for example (later to control the PLO and the Palestinian Authority following Oslo II in 1995), conducted its inaugural attack in January 1965, targeting an Israeli water pump that moved water from the wetter Galilee region towards the Negev desert in the south. The attack failed, with the explosives failing to detonate and the operatives being captured upon their return to Lebanon by Arab authorities. However, the attacks ramped up with particular intensity in the 1965-67 period. Many of these attacks also led to skirmishes, with Israel holding Arab states responsible for attacks originating in their territory. Palestinian groups, for their part, were hoping to spark a war, believing that if they could force a war, then they could force the Arab states to finally wipe Israel out, once and for all. While Arab leaders were more hesitant and believed they weren't ready for that kind of fight yet (though they certainly would've loved to be), the Palestinian groups were quite confident in their desires, and some Arab states viewed them as good proxies for "letting off steam" among Palestinian refugees within their borders, and also as a good way to bleed Israel and its populace.

There are a couple of important points to note in that regard. One is Samu. In November 1966, an Israeli border police vehicle (on the Israeli side of the border) hit a mine, killing three police officers. Israel, at this point rather fed up with the skirmishes, decided to launch a bigger display of force against villages in the West Bank that it believed had aided Fatah in carrying out the attack. King Hussein of Jordan had sent a personal apology to the American ambassador, requesting that he pass it to the Israelis. However, the ambassador went home for the weekend without passing it on, so Israel only learned of it after it had responded. Israel sent 400 men, 10 tanks, and 40 half-tracks to attack the village of Samu (as well as Rujm al-Madfa, where they demolished the police station and moved on). They hoped this would deter future fighting, but it did the opposite. Israeli forces destroyed much of the village, but a patrol of Jordanian troops began to approach, and since they had no time to escape, the Israeli forces laid an ambush. The fight became way bigger than Israel had wanted, including an air battle, the death of the Israeli battalion commander, and Israel was censured at the UN Security Council. 15 Jordanian troops were killed, 3 Arab civilians died, and riots broke out demanding King Hussein's overthrow. Israel had intended to get the villages to demand Hussein do more to prevent Fatah attacks and protect them. Instead, riots demanding he step down began in Hebron, Jerusalem, and Bethlehem, and Jordanian forces eventually opened fire, killing 4. The constant raids into Israel didn't peter out; Palestinians believed they were getting closer to starting the war they wanted, so they continued operations targeting civilians, hoping to spark another Samu that would spiral into a war. It also gave them the impression that Arab forces were indeed stronger than they seemed, and could defeat Israel.

This backfired operation helped set the backdrop for an event later on, in Syria. Syria was one of the most open backers of the Palestinian guerrilla groups. In Egypt and Jordan, the support was sporadic at best, and only for certain groups, so the United States was quite resistant and critical when Israel retaliated. On the Syrian front, however, the support was unabashed, leading one Defense Department official to actually ask Israel why it wasn't answering more forcefully, asking "Why the hell didn't you beat [the Syrians] over the head when it would have been the most natural thing to do?" On the Syrian front, Israel was preparing for a Samu-like reprisal strike, but it hoped to prevent it from escalating in the same way by being more precise in how it used force. On March 31, 1967, the opportunity presented itself, with Palestinian groups planting explosives under a water pump and railroad tracks. Israel decided to provoke the Syrians into attacking Israeli forces, so that there would be no doubt it had the right to respond. Israel sent tractors into the demilitarized zone, the Syrians fired on the tractors, but things were a bit different on April 7. Normally small arms fire was used, and no one ended up hurt. This time, the Syrians used cannons, so Israel fired back with its tanks, and the Syrian guns fired on Israeli villages across the border. Israel finally had the chance to respond that it was looking for, without anyone worrying that it was overreacting to Palestinian attacks. After an hour of Syrian shelling, despite UN attempts to mediate a quick ceasefire, Israel activated its air force, because Syria said it would only stop if Israel stopped agricultural work in the DMZ, which Israel refused to do (Syria also refused, for that matter, to restrict DMZ activities). Israel had been reluctant to take that step, believing a ceasefire after mutual shelling would be good enough to establish deterrence and convince the Syrians to stop sponsoring Palestinian attacks on Israelis. However, Israel's decision came because the intensified artillery barrage showed no signs of stopping. The Israeli jets bombed Syrian targets, villages and military alike, putting an end to the shelling. However, the Syrians also activated their MiGs. Israel's jets were French Mirage fighters, heavily modified to be more effective. The Israelis fought Syrian MiG-21s supplied by the Soviet Union. They downed two MiGs over the Golan Heights, and another 4 were downed in a dogfight over Damascus, the Syrian capital. The capital watched as the Syrians reported "downing" 5 Israeli planes, when in reality, Syria had lost 6 while Israel lost none. The Israeli pilots were far better trained, and Syria's air force was often terrible by comparison. The events of April 7 gave Syria a black eye, and set the backdrop for the later events leading up to the war. Israel was tired of launching these retaliations and believed it had to make a show of force to deter them, but the Arab states were unwilling or unable to stop the Palestinian guerrillas, were themselves tired of fighting over the DMZ, and were on edge because they saw the escalations as a slide towards war, as did the Israelis. It's for that reason that many sometimes describe the events of May 1967 as both sides "bumbling" into a war. It's actually a bit of a case study in how controlled escalation can go wrong, according to some.

Continued below in a response ot my own comment.