In the chart below, I'm showing the rolling correlations between stock returns and bond returns. (The relationship would be flipped if you are studying stock returns vs interest rates).
As you can see, for the bulk of the history since 1960s, bond returns and stock returns were indeed positively correlated; i.e., when stocks went up, bonds went up too (and interest rate declined). The past 15 years or so, however, bond returns and stock returns have turned negative (i.e., stocks go up, bonds go down/interest rate goes up).
The historical "norm" was that negative supply shocks tended to cause the equity market to tank, while the coinciding high inflation hurt bonds as well. Nowadays, with inflation expectation very well anchored, bonds have largely become synonymous with recession-hedges – people buy bonds (cause rates to decline) when stocks are performing poorly ("flight-to-quality").
The past few years' experience is also interesting. The Fed buying up a lot of Treasuries (quantitative easing) plus the zero-interest-rate-policy has emboldened risk taking, sending S&P to records.
So to sum up, how stocks & interest rates move relative to each other is very much dependent on the macro backdrop.
Reference: Antii Ilmanen's "Expected Returns" (2011)