The manner in which 'third generation' Mexican-Americans are identified, predicated on self-reported ethnic identity rather than grandparental nativity, is imprecise and potentially confounded with later generations. This study examines the degree to which this imprecision accounts for the stagnation found in many past studies of integration, including fertility integration. We use data from the study Immigration and Intergenerational Mobility in Metropolitan Los Angeles (IIMMLA), which includes questions on grandparental nativity, to construct a better-defined third generation. When IIMMLA data are aggregated into the commonly measured third-plus generation, the findings show a pattern of slow intergenerational-fertility integration, similar to previous research using the Current Population Survey or other data with third-plus generation measures. However, when we consider a third generation identified by grandparental nativity, evidence emerges of a faster fertility decline, bringing the third generation's fertility more in line with the native-born population as a whole. The results suggest that conclusions about integration based on third-plus generation measures should be regarded with greater scepticism and support efforts to include in surveys questions on grandparental nativity to enable the isolation of a genuine third generation among ethnic groups with immigrant origins.