Memory hasn't always been the limitation. IBM identified (and in many ways, encouraged the adoption of) two kinds of computing: scientific and data processing. Data processing computers (such as the IBM 702 from 1953) had relatively modest processor power, but enough memory to hold all the parameters for the day's business transactions to ensure that processing could be finished before the next day's work was needed. Compare this to the scientific machines (such as 1954's 704) that had tiny amounts of memory but could crunch floating point numbers quickly for numerical analysis. IBM's business and scientific machines ran completely different instruction sets, and evolved into two completely separate computer cultures.
Since you asked for anecdotes, my father was once (briefly) accused of breaking a utility's billing mainframe. Dad was an analyst for ICT in the early 1960s, and his team was asked to spec and write a network flow analysis for the Scottish Gas Board's pipes. The data (which would likely fit on a single screen of spreadsheet today) was too big to analyze in any of ICT's scientific machines, so they had to write it for the utility's billing computer. Even with the “large” (a few kilo-words) amount of memory in the billing machine, the task had to be broken into many processing chunks, with each chunk dumping its data and chaining in the next task. The main analysis chunk was estimated to just fit in the processing time that the Billing Department had grudgingly allowed the pipeline boffins on the commercial machine.
You may not remember, but in the days of mag tape drives (✇✇), computer data processing activity correlated with the tapes spinning now and again. Each data record was read, processed and written, and the tapes advanced a bit every time. A working computer had mag tapes that moved; everyone knew that. Even poets knew it: Brautigan wrote of a techno-pastoral utopia “where deer stroll peacefully / past computers / as if they were flowers / with spinning blossoms”. So spinning tapes = working computer = happy operator.
Dad's network analysis job had to be run at an unpleasant hour of the night, after the day's billing had been run but before the next morning's readings started to come in. The job got set up, and the accounting operator approved of the first stages: the tapes kept moving, so the computer was working.
Then came the main analysis. Once the tape drives had loaded all the data into core, the slow processor in the accounts machine started its work. The tapes stopped … and stayed stopped. After a couple of minutes, the operator started to fret, asking if it was supposed to do this. Despite positive assurances, his fretting increased as the minutes ticked by with no tape activity. Apparently the operator was almost screaming, threatening to have my dad escorted from the building and his company billed for the whole computer that they'd “broken”, when a tape moved! The results spooled to tape from core, and the operator calmed down a bit (as did my dad, who was apparently beginning to have doubts of his own.) I don't know how long the job took to run — probably less than half an hour — but it certainly pushed that computer operator's experience well out of the comfort zone.
Personally, I've also experienced processor limitations (making Julia Set animations on an Amiga A500 in the early 1990s, the first time I had to let a computer run all night) and I/O bandwidth bottlenecks (ripping > 1000 CDs on two computers and four CD-ROM drives: it took a couple of weeks). But none have resulted in threats, so far at least.