The story most people have been telling about the candidacy of Hillary Clinton is this: Once the dominant figure in the Democratic primary race, she's seen her approval ratings slide -- thanks largely to her response to the State Department e-mail issue. That's created space for Sen. Bernie Sanders (I-Vt.) to claw into contention, as he continues to eat into her lead.

That story has some validity buried within it, and polls in Iowa and New Hampshire do show a tighter-than-anticipated race in the first two states, with Clinton edging Sanders in the former and Sanders edging Clinton in the latter.

But a new Monmouth University poll shows that this narrative has also got some major holes.

In the new national survey, the percentage of Democrats who indicate they plan to vote for Clinton sinks to 42 percent -- 18 points lower than the support she saw in April, and only 20 points above the second-place contender. That's a bigger deal; the previous smallest gap was 34 points in July.

But the reason that Clinton fell 10 points isn't because Democrats are viewing her less favorably and turning to Sanders. Instead, they're responding to the recent Joe Biden boomlet. Between Monmouth's August and September polls, Biden has gained 10 points -- exactly what Clinton lost.

That's reinforced in a look at their favorability ratings, too. While Clinton's favorability rating with Democrats has slipped, it's mostly her favorability rating with all voters that's really taken a beating. She is still viewed favorably by 71 percent of Democrats -- exactly the same percentage who view Biden favorably. The percentage viewing her unfavorably is higher, yes, but only by five points.

Overall, their net favorability ratings -- those viewing them positively minus those viewing them negatively -- are pretty even. (Sanders, being much less well-known, has lower percentages who view him positively or negatively.)

We've noted before that Biden and Clinton share a base. Monmouth's poll finds that 41 percent of non-Biden voters would be "very" or "somewhat" likely to support him if he were to decide to run (a number that's very close to Clinton's 42 percent support, though we don't have crosstabs to show how many Clinton backers would consider Biden). If Biden doesn't decide to run, other polling suggests that it's Clinton who will benefit from his not doing so -- just as she's the one who suffers when his candidacy is polled.

In other words, this poll offers a different story. Hillary Clinton's base of support isn't rock solid, and a Biden run could hurt her. But by "hurt her," we mean drop her support under 50 percent -- and who knows how long that would last. Biden's popularity is in large part because he isn't a candidate. Once he gets in, it stands to reason that he would see the same dip in favorability that dragged Hillary Clinton all the way down to -- Joe Biden levels.