It depends on what you mean by a "long string of low level wins" and a "large" jackpot. There are several ways to interpret your question, and I will give you just one based on recent events.
Between April 21 and August 1 of this year, we had a string of runs in which the jackpot never passed $100M in advertised annuity value. The lengths of the runs in this period were 8 (about average) followed by below average length runs of 5, 4, 3, 6 and 4 draws.
Working from my modeling program, I calculate that the probability of having a set exactly consisting of runs with this many drawings (in any order) is 0.11%. This is smaller than the probability of a jackpot rolling beyond $600M annuity from a given first draw. Thus the events of this summer were improbable.
However, we should recognize that there are many other sequences that might constitute a "string of low level wins." I don't have the time to examine a subset of them, never mind all of the different possible ways in which such a string might occur.
It is easier to ask the question, how long is it likely to take to have a jackpot that is more than $500M in advertised annuity? Keep in mind that such a thing can never become a certainty, but if one looks at a longer and longer stretch of time, the probability of such a jackpot becomes more and more likely.
Here is a table that indicates a percentage probability of a $500M jackpot and the length of time indicated in draws, weeks, and years that such a jackpot will have the probability indicated. To convert from draws to weeks, I have used the average value of 7.47 draws/run.
(I'm sorry the header ran together, but it's an unavoidable consequence of the cut and paste from excel.)
Odds of $500M jackpot occurring in a period | Number of runs required to produce this probability of a $500M jackpot | Number of draws required to reach these odds | Number of weeks required to reach these odds | Number of years required to reach these odds |
5.00% | 6 | 43 | 22 | 0.42 |
10.00% | 12 | 89 | 45 | 0.86 |
15.00% | 18 | 137 | 69 | 1.32 |
20.00% | 25 | 189 | 94 | 1.81 |
25.00% | 33 | 243 | 122 | 2.34 |
30.00% | 40 | 302 | 151 | 2.90 |
35.00% | 49 | 364 | 182 | 3.50 |
40.00% | 58 | 432 | 216 | 4.15 |
45.00% | 68 | 506 | 253 | 4.86 |
50.00% | 78 | 586 | 293 | 5.64 |
55.00% | 90 | 675 | 338 | 6.49 |
60.00% | 104 | 775 | 387 | 7.45 |
65.00% | 119 | 888 | 444 | 8.54 |
70.00% | 136 | 1018 | 509 | 9.79 |
75.00% | 157 | 1172 | 586 | 11.27 |
80.00% | 182 | 1361 | 680 | 13.09 |
85.00% | 215 | 1604 | 802 | 15.42 |
90.00% | 261 | 1947 | 973 | 18.72 |
95.00% | 339 | 2533 | 1267 | 24.36 |
This event could occur on the current run, or not at all, but it likely to happen within the next six years. There's a five percent chance we'll see such a jackpot this year. But t have a 95% chance, we would need to wait almost 25 years.
All these events are random. There is no way to accurately predict exactly when it will happen, but we can get an idea of how likely it is in a given period.