There was indeed an outburst of enthusiasm for tulips in the Netherlands in 1636-1637, with associated high prices. This has long been claimed as the first well-documented financial bubble, though whether it actually was a bubble has recently been debated. 

Full disclosure, I haven't watched the movie and I'm not a biologist, let alone a botanist. My interest in this is from the financial history point of view.

What happened?

Tulips arrived in Western Europe from Turkey during the 16th century, and between their novelty and beauty, a craze broke out for them amongst the wealthy. Particularly prized were tulips with dramatic stripes. Said stripes were/are mainly caused by a virus (identified in the 20th century), not genes, so this made breeding the most prized tulips difficult; bulbs might produce the pattern one year but not the next. Prices of the bulbs rose in some cases to very high levels. The most valuable tulip of the 1620s, the Sempter Augustus, was in the hands of a single owner who held onto it even as prices offered reached 3000 gulden per bulb in 1625 - about 10x the earnings of a carpenter at the time. 

Note that I'm here talking about bulbs, not flowers or seeds. Bulbs produce offshoots underground each year, called 'offshoots', which grow into new bulbs, which produce new offsets, etc. The bulbs were important as they are clones, unlike seeds, so if you want to have the same flower again next year you want a bulb (and I am totally drawing on my sources for this info.)

Tulip growers responded to the enthusiasm by breeding multiple varities, but supplies were restricted as new bulbs had to grow for a number of years before they bloomed for the first time. So supply was restricted, and the underlying value of the bulb was uncertain.

Interest in bulbs was particularly noted in the Netherlands, where incomes were high, the Dutch being probably the richest nation per capita in Europe at this time. Merchant trade and financial trading was well advanced, so people had the income to spend large sums of money on bulbs, or, as in the case of the Semper Augustus, at least offer large sums. 

In 1635, there were two financial innovations in the Dutch tulip trade: moving from sale-by-the-bulb to sale-by-bulb weight - a smaller bulb was less mature and would take longer to start producing flowers. The next innovation was the bulb promissory note, a note promising to provide a bulb in the future. It first appeared in 1635, but only became common in 1636. So people might sell bulbs they didn't actually have in their possession. These two innovations came as more people started to move into the market, expanding it from tulip enthusiasts to speculators in it not for the tulips but for the money.

Bulb prommissory notes would be traded where neither party would expect to have a bulb when the contract was settled, instead it was expected that one would pay the other the difference between the contract and settlement price when the contract finished. However, unlike modern futures trading, the contracts weren't continuously marked-to-market, nor were they backed by exchanges, nor were margins required. Oh, and a lot of tulip trading took place in taverns, where drinks were readibly available.  Alcoholic drinks. 

All in all, these conditions were very favourable for a market bubble. Which is what happened. From October 1636, prices started to really take off, over the next 3 months some individual bulbs rose ten-fold, some prices for bulk quantities of cheaper bulbs increased 20 fold.

And then the first bulb sprouts began to appear in the first week of February 1637. Suddenly much information became available about the future supply of bulbs that wasn't a week before. Prices crashed over February 3rd to February 7th (this was in the days of travel by horse back).

A series of three pamphlets were produced at the time critiquing the tulip craze, they were all published by Adriaen Roman in Haarlem, in 1637. These moralise against the trade, and also include some prices on tulips, though specialist economic historians aren't confident in their accuracy. They're not the only source of data, but they're a commonly used one.

What was the economic impact?

Not much.  The local authorities, following a Court of Holland ruling on 25 April 1637, imposed a moratorium on any official contract enforcement, so bulb traders were left to negotiate agreements about themselves. The Dutch merchant class was relatively small, and had strong social linkages with each otherm and most contract disputes appear to have been accepted more or less amicably, though it ws a shock to norms. Almost all tulip traders had other careers, and hadn't given up their day jobs, and most were comparatively wealthy, at least the ones who appear in the records. Which is not to say that definitely no one went bankrupt, some people may have, but it wasn't a massive economic shock. We don't have anything like modern economic statistics back for the 17th century but there's no sign of a general recession.

Was it a bubble?

A common definition of a financial bubble in the academic literature is “a situation in which temporarily high prices are sustained largely by investors’ enthusiasm rather than by consistent estimation of real value”. This makes intuitive sense but raises questions about what is 'real value'. It's not just a matter of how things eventually turn out: if you buy fire insurance at the start of the year and your house doesn't burn down over that year that doesn't mean that the insurance didn't offer real value. So debates about bubbles tend to be not so much about the price rise itself but was it justified given information available to the participants at the time.

Starting in the 1980s, some authors began to question whether there really was a finacial bubble, particularly Peter Garber, who in 1989 argued that other flowering bulbs displayed similar pricing patterns (except maybe the January runup) and this can be explained by competitive models with limited information. Earl Thompson, 2006, disagreed on the basis that the 1637 price fall was much more dramatic than Garber's other examples and argued that the high prices were the result of a retrospective contract change made by Dutch officials. McClure & Thomas (2017) disagree with Thompson, arguing he was building a theory only on 3 observations, misinterpreting 1 of them and ignoring other evidence. They argue it was a bubble caused by the absence of information when the bulbs were in the ground, thus the timing of the price bust.

So current histography, it probably was a bubble, but there is a bit of doubt.

Sources

James McClure, David Thomas, Explaining the timing of tulipmania’s boom and bust: historical context, sequestered capital and market signals Financial History Review 24.2, (2017), pp 121-141, https://www.cambridge.org/core/services/aop-cambridge-core/content/view/20BEB345A7BB4BF2E84C07F9077361A1/S0968565017000154a.pdf/explaining_the_timing_of_tulipmanias_boom_and_bust_historical_context_sequestered_capital_and_market_signals.pdf

A. Maurits van der Veen (2012) The Dutch Tulip Mania: The Social Foundations of a Financial Bubble,  College of William & Mary, October 2012, http://www.maurits.net/Research/TulipMania.pdf

https://eh.net/book_reviews/tulipmania-money-honor-and-knowledge-in-the-dutch-golden-age/

Earl Thompson, 2006, The Tulipmania: fact or artifact?, Public Choice