Tabs have become integral to browsing the web yet have changed little since their introduction nearly 20 years ago. In contrast, the internet has gone through dramatic changes, with tasks increasingly moving from the desktop to the cloud. In this paper, we investigate how tabs support the diverse functionalities of the modern web. Based on interviews and surveys, we found that many users struggle to manage and switch between streams of complex tasks, resulting in significant breakdowns. We discover competing pressures pushing for the opening of tabs (such as revisitation costs, sunk costs, uncertainty around relevance, and mismatches between the aspirational vs. actual self) versus pushing for closing tabs (such as limited attention and resources). Our results indicate these pressures lead to issues ranging from tab overload, hoarding obsolete tabs, frustration, and even shamefulness. Based on our findings, we develop implications for designing better ways for users to interact with the modern web.