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In reliability and life data analysis, the Weibull distribution is widely used to accommodate more data characteristics by changing the values of the parameters. We frequently observe many zeros or close to zero data points in reliability and life testing experiments. We call this phenomenon a nearly instantaneous failure. Many researchers modified the commonly used univariate parametric models such as exponential, gamma, Weibull, and log-normal distributions to appropriately fit such data having instantaneous failure observations. Researchers also find bivariate correlated life testing data having many observations near a particular point while the remaining observations follow some continuous distribution. This situation defines as responses having early failures for such bivariate responses. If the point is the origin, then we call the situation a nearly instantaneous failure for the responses. Here, we propose a modified bivariate Weibull distribution that allows early failure by combining bivariate uniform distribution and bivariate Weibull distribution. The bivariate Weibull distribution is constructed using a 2-dimensional copula, assuming the marginal distributions as two parametric Weibull distributions. We derive some properties of that modified bivariate Weibull distribution, mainly the joint probability density function, the survival (reliability) function, and the hazard (failure rate) function. The model's unknown parameters are estimated using the Maximum Likelihood Estimation (MLE) technique combined with a machine learning clustering algorithm. Numerical examples are provided using simulated data to illustrate and test the performance of the proposed methodologies. The method is also applied to real data and compared with existing approaches to model such data in the literature.